#[allow(unused_imports)]
use proconio::{
    fastout, input,
    marker::{Bytes, Chars, Isize1, Usize1},
};

fn main() {
    println!("ライブラリ用のダミーコードです。");
}

//https://github.com/rust-lang-ja/ac-library-rs
//https://github.com/manta1130/competitive-template-rs

use complex::*;
use convolution::*;
use counter::*;
use dlp::*;
use dpfactorial::*;
use dsu::*;
use dsu_potential::*;
use eratosthenes::*;
use fenwicktree::*;
use fft::*;
use geometry::*;
use global::*;
use graph::*;
use graph_search::*;
use input::*;
use lazysegtree::*;
use lca::*;
use math::*;
use matrix::*;
use maxflow::*;
use mex::*;
use mincostflow::*;
use modint::*;
use ordfloat::*;
use pcg64::*;
use primenumber::*;
use raq_bit::*;
use rng_trait::*;
use rollinghash::*;
use scc::*;
use segtree::*;
use segtree_template::*;
use string::*;
use timer::*;
use twosat::*;
use vectools::*;
use xorshift::*;

pub mod complex {
    #![warn(missing_docs)]

    use std::fmt;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};

    #[derive(Clone, PartialEq, Debug, Eq, Copy, Default)]
    pub struct Complex<T> {
        pub re: T,
        pub im: T,
    }

    impl<T> Complex<T> {
        #[must_use]
        pub const fn new(re: T, im: T) -> Self {
            Self { re, im }
        }
    }

    impl<T: Copy + Neg<Output = T>> Complex<T> {
        #[must_use]
        pub fn conj(self) -> Self {
            Self::new(self.re, -self.im)
        }
    }

    impl<T> Complex<T>
    where
        T: Copy + Add<Output = T> + Mul<Output = T>,
    {
        #[must_use]
        pub fn norm_sqr(self) -> T {
            self.re * self.re + self.im * self.im
        }
    }

    impl<T> Complex<T>
    where
        T: Copy + Add<Output = T> + Mul<Output = T> + Into<f64>,
    {
        #[must_use]
        pub fn norm(self) -> f64 {
            self.norm_sqr().into().sqrt()
        }

        #[must_use]
        pub fn arg(self) -> f64 {
            self.im.into().atan2(self.re.into())
        }
    }

    impl<T> From<(T, T)> for Complex<T> {
        fn from((re, im): (T, T)) -> Self {
            Self::new(re, im)
        }
    }

    impl<T: fmt::Display> fmt::Display for Complex<T> {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "{}{:+}i", self.re, self.im)
        }
    }

    impl<T: Add<Output = T>> Add for Complex<T> {
        type Output = Self;
        fn add(self, rhs: Self) -> Self {
            Self::new(self.re + rhs.re, self.im + rhs.im)
        }
    }
    impl<T: AddAssign> AddAssign for Complex<T> {
        fn add_assign(&mut self, rhs: Self) {
            self.re += rhs.re;
            self.im += rhs.im;
        }
    }
    impl<T: Sub<Output = T>> Sub for Complex<T> {
        type Output = Self;
        fn sub(self, rhs: Self) -> Self {
            Self::new(self.re - rhs.re, self.im - rhs.im)
        }
    }
    impl<T: SubAssign> SubAssign for Complex<T> {
        fn sub_assign(&mut self, rhs: Self) {
            self.re -= rhs.re;
            self.im -= rhs.im;
        }
    }
    impl<T> Mul for Complex<T>
    where
        T: Copy + Add<Output = T> + Sub<Output = T> + Mul<Output = T>,
    {
        type Output = Self;
        fn mul(self, rhs: Self) -> Self {
            Self::new(
                self.re * rhs.re - self.im * rhs.im,
                self.re * rhs.im + self.im * rhs.re,
            )
        }
    }
    impl<T> MulAssign for Complex<T>
    where
        T: Copy + Add<Output = T> + Sub<Output = T> + Mul<Output = T>,
    {
        fn mul_assign(&mut self, rhs: Self) {
            *self = *self * rhs;
        }
    }
    impl<T> Div for Complex<T>
    where
        T: Copy
            + Add<Output = T>
            + Sub<Output = T>
            + Mul<Output = T>
            + Div<Output = T>
            + Neg<Output = T>,
    {
        type Output = Self;
        fn div(self, rhs: Self) -> Self {
            let denominator = rhs.norm_sqr();
            let numerator = self * rhs.conj();
            Self::new(numerator.re / denominator, numerator.im / denominator)
        }
    }
    impl<T> DivAssign for Complex<T>
    where
        T: Copy
            + Add<Output = T>
            + Sub<Output = T>
            + Mul<Output = T>
            + Div<Output = T>
            + Neg<Output = T>,
    {
        fn div_assign(&mut self, rhs: Self) {
            *self = *self / rhs;
        }
    }
}
pub mod convolution {

    macro_rules! modulus {
    ($($name:ident),*) => {
        $(
            #[derive(Copy, Clone, Eq, PartialEq)]
            enum $name {}

            impl Modulus for $name {
                const VALUE: u32 = $name as _;
                const HINT_VALUE_IS_PRIME: bool = true;

                fn butterfly_cache() -> &'static ::std::thread::LocalKey<::std::cell::RefCell<::std::option::Option<$crate::modint::ButterflyCache<Self>>>> {
                    thread_local! {
                        static BUTTERFLY_CACHE: ::std::cell::RefCell<::std::option::Option<$crate::modint::ButterflyCache<$name>>> = ::std::default::Default::default();
                    }
                    &BUTTERFLY_CACHE
                }
            }
        )*
    };
}

    use crate::{
        internal_bit, internal_math,
        modint::{ButterflyCache, Modulus, RemEuclidU32, StaticModInt},
    };
    use std::{
        cmp,
        convert::{TryFrom, TryInto as _},
        fmt,
    };

    #[allow(clippy::many_single_char_names)]
    pub fn convolution<M>(a: &[StaticModInt<M>], b: &[StaticModInt<M>]) -> Vec<StaticModInt<M>>
    where
        M: Modulus,
    {
        if a.is_empty() || b.is_empty() {
            return vec![];
        }
        let (n, m) = (a.len(), b.len());

        if cmp::min(n, m) <= 60 {
            let (n, m, a, b) = if n < m { (m, n, b, a) } else { (n, m, a, b) };
            let mut ans = vec![StaticModInt::new(0); n + m - 1];
            for i in 0..n {
                for j in 0..m {
                    ans[i + j] += a[i] * b[j];
                }
            }
            return ans;
        }

        let (mut a, mut b) = (a.to_owned(), b.to_owned());
        let z = 1 << internal_bit::ceil_pow2((n + m - 1) as _);
        a.resize(z, StaticModInt::raw(0));
        butterfly(&mut a);
        b.resize(z, StaticModInt::raw(0));
        butterfly(&mut b);
        for (a, b) in a.iter_mut().zip(&b) {
            *a *= b;
        }
        butterfly_inv(&mut a);
        a.resize(n + m - 1, StaticModInt::raw(0));
        let iz = StaticModInt::new(z).inv();
        for a in &mut a {
            *a *= iz;
        }
        a
    }

    pub fn convolution_raw<T, M>(a: &[T], b: &[T]) -> Vec<T>
    where
        T: RemEuclidU32 + TryFrom<u32> + Clone,
        T::Error: fmt::Debug,
        M: Modulus,
    {
        let a = a.iter().cloned().map(Into::into).collect::<Vec<_>>();
        let b = b.iter().cloned().map(Into::into).collect::<Vec<_>>();
        convolution::<M>(&a, &b)
            .into_iter()
            .map(|z| {
                z.val()
                    .try_into()
                    .expect("the numeric type is smaller than the modulus")
            })
            .collect()
    }

    #[allow(clippy::many_single_char_names)]
    pub fn convolution_i64(a: &[i64], b: &[i64]) -> Vec<i64> {
        const M1: u64 = 754_974_721; // 2^24
        const M2: u64 = 167_772_161; // 2^25
        const M3: u64 = 469_762_049; // 2^26
        const M2M3: u64 = M2 * M3;
        const M1M3: u64 = M1 * M3;
        const M1M2: u64 = M1 * M2;
        const M1M2M3: u64 = M1M2.wrapping_mul(M3);

        modulus!(M1, M2, M3);

        if a.is_empty() || b.is_empty() {
            return vec![];
        }

        let (_, i1) = internal_math::inv_gcd(M2M3 as _, M1 as _);
        let (_, i2) = internal_math::inv_gcd(M1M3 as _, M2 as _);
        let (_, i3) = internal_math::inv_gcd(M1M2 as _, M3 as _);

        let c1 = convolution_raw::<i64, M1>(a, b);
        let c2 = convolution_raw::<i64, M2>(a, b);
        let c3 = convolution_raw::<i64, M3>(a, b);

        c1.into_iter()
            .zip(c2)
            .zip(c3)
            .map(|((c1, c2), c3)| {
                const OFFSET: &[u64] = &[0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3];

                let mut x = [(c1, i1, M1, M2M3), (c2, i2, M2, M1M3), (c3, i3, M3, M1M2)]
                    .iter()
                    .map(|&(c, i, m1, m2)| {
                        c.wrapping_mul(i).rem_euclid(m1 as _).wrapping_mul(m2 as _)
                    })
                    .fold(0, i64::wrapping_add);

                let mut diff = c1 - internal_math::safe_mod(x, M1 as _);
                if diff < 0 {
                    diff += M1 as i64;
                }
                x = x.wrapping_sub(OFFSET[diff.rem_euclid(5) as usize] as _);
                x
            })
            .collect()
    }

    #[allow(clippy::many_single_char_names)]
    fn butterfly<M: Modulus>(a: &mut [StaticModInt<M>]) {
        let n = a.len();
        let h = internal_bit::ceil_pow2(n as u32);

        M::butterfly_cache().with(|cache| {
            let mut cache = cache.borrow_mut();
            let ButterflyCache { sum_e, .. } = cache.get_or_insert_with(prepare);
            for ph in 1..=h {
                let w = 1 << (ph - 1);
                let p = 1 << (h - ph);
                let mut now = StaticModInt::<M>::new(1);
                for s in 0..w {
                    let offset = s << (h - ph + 1);
                    for i in 0..p {
                        let l = a[i + offset];
                        let r = a[i + offset + p] * now;
                        a[i + offset] = l + r;
                        a[i + offset + p] = l - r;
                    }
                    now *= sum_e[(!s).trailing_zeros() as usize];
                }
            }
        });
    }

    #[allow(clippy::many_single_char_names)]
    fn butterfly_inv<M: Modulus>(a: &mut [StaticModInt<M>]) {
        let n = a.len();
        let h = internal_bit::ceil_pow2(n as u32);

        M::butterfly_cache().with(|cache| {
            let mut cache = cache.borrow_mut();
            let ButterflyCache { sum_ie, .. } = cache.get_or_insert_with(prepare);
            for ph in (1..=h).rev() {
                let w = 1 << (ph - 1);
                let p = 1 << (h - ph);
                let mut inow = StaticModInt::<M>::new(1);
                for s in 0..w {
                    let offset = s << (h - ph + 1);
                    for i in 0..p {
                        let l = a[i + offset];
                        let r = a[i + offset + p];
                        a[i + offset] = l + r;
                        a[i + offset + p] = StaticModInt::new(M::VALUE + l.val() - r.val()) * inow;
                    }
                    inow *= sum_ie[(!s).trailing_zeros() as usize];
                }
            }
        });
    }

    fn prepare<M: Modulus>() -> ButterflyCache<M> {
        let g = StaticModInt::<M>::raw(internal_math::primitive_root(M::VALUE as i32) as u32);
        let mut es = [StaticModInt::<M>::raw(0); 30]; // es[i]^(2^(2+i)) == 1
        let mut ies = [StaticModInt::<M>::raw(0); 30];
        let cnt2 = (M::VALUE - 1).trailing_zeros() as usize;
        let mut e = g.pow(((M::VALUE - 1) >> cnt2).into());
        let mut ie = e.inv();
        for i in (2..=cnt2).rev() {
            es[i - 2] = e;
            ies[i - 2] = ie;
            e *= e;
            ie *= ie;
        }
        let sum_e = es
            .iter()
            .scan(StaticModInt::new(1), |acc, e| {
                *acc *= e;
                Some(*acc)
            })
            .collect();
        let sum_ie = ies
            .iter()
            .scan(StaticModInt::new(1), |acc, ie| {
                *acc *= ie;
                Some(*acc)
            })
            .collect();
        ButterflyCache { sum_e, sum_ie }
    }
}
pub mod counter {
    #![warn(missing_docs)]

    use std::collections::HashMap;
    use std::hash::Hash;

    pub trait Counter: Iterator + Sized {
        fn counter(self) -> HashMap<Self::Item, usize>
        where
            Self::Item: Eq + Hash,
        {
            let mut result = HashMap::new();
            for item in self {
                *result.entry(item).or_default() += 1;
            }
            result
        }
    }
    impl<I: Iterator> Counter for I {}
}
pub mod dlp {
    #![warn(missing_docs)]

    use std::collections::HashMap;

    fn gcd(mut a: u64, mut b: u64) -> u64 {
        while b != 0 {
            (a, b) = (b, a % b);
        }
        a
    }

    fn mul_mod(a: u64, b: u64, modulus: u64) -> u64 {
        (u128::from(a) * u128::from(b) % u128::from(modulus)) as u64
    }

    fn pow_mod(mut base: u64, mut exponent: u64, modulus: u64) -> u64 {
        let mut result = 1 % modulus;
        while exponent != 0 {
            if exponent & 1 != 0 {
                result = mul_mod(result, base, modulus);
            }
            base = mul_mod(base, base, modulus);
            exponent >>= 1;
        }
        result
    }

    fn inverse_mod(value: u64, modulus: u64) -> u64 {
        fn egcd(a: i128, b: i128) -> (i128, i128, i128) {
            if b == 0 {
                (a, 1, 0)
            } else {
                let (g, x, y) = egcd(b, a % b);
                (g, y, x - (a / b) * y)
            }
        }
        let (g, x, _) = egcd(i128::from(value), i128::from(modulus));
        assert_eq!(g, 1);
        x.rem_euclid(i128::from(modulus)) as u64
    }

    fn bsgs_coprime(base: u64, target: u64, modulus: u64, positive: bool) -> Option<u64> {
        if gcd(target, modulus) != 1 {
            return None;
        }
        let block = modulus.isqrt().saturating_add(1);
        let mut babies = HashMap::new();
        let (first, last) = if positive { (1, block) } else { (0, block - 1) };
        let mut value = pow_mod(base, first, modulus);
        for exponent in first..=last {
            if value == target {
                return Some(exponent);
            }
            if babies.insert(value, exponent).is_some() {
                return None;
            }
            value = mul_mod(value, base, modulus);
        }
        let giant = pow_mod(inverse_mod(base, modulus), block, modulus);
        let mut current = target;
        for i in 0..=block {
            if let Some(&j) = babies.get(&current) {
                return i.checked_mul(block)?.checked_add(j);
            }
            current = mul_mod(current, giant, modulus);
        }
        None
    }

    #[must_use]
    pub fn dlp(base: u64, target: u64, modulus: u64, allow_zero: bool) -> Option<u64> {
        assert_ne!(modulus, 0, "the modulus must be positive");
        if modulus == 1 {
            return Some(u64::from(!allow_zero));
        }
        let base = base % modulus;
        let mut target = target % modulus;
        let mut reduced_modulus = modulus;
        let mut accumulated = 1_u64;
        let mut removed = 0_u64;

        loop {
            let divisor = gcd(base, reduced_modulus);
            if divisor == 1 {
                break;
            }
            if target == accumulated && (allow_zero || removed > 0) {
                return Some(removed);
            }
            if !target.is_multiple_of(divisor) {
                return None;
            }
            target /= divisor;
            reduced_modulus /= divisor;
            accumulated = mul_mod(accumulated, base / divisor, reduced_modulus);
            removed += 1;
        }

        let adjusted = mul_mod(
            target,
            inverse_mod(accumulated, reduced_modulus),
            reduced_modulus,
        );
        let positive = !allow_zero && removed == 0;
        bsgs_coprime(base % reduced_modulus, adjusted, reduced_modulus, positive)
            .and_then(|x| x.checked_add(removed))
    }
}
pub mod dpfactorial {
    #![warn(missing_docs)]

    use crate::modint::{ModInt, ModIntBase};

    #[derive(Clone, Debug)]
    pub struct DpFactorial<M: ModIntBase = ModInt> {
        factorials: Vec<M>,
        inverse_factorials: Vec<M>,
        modulus: u32,
    }

    impl<M: ModIntBase> Default for DpFactorial<M> {
        fn default() -> Self {
            Self::new()
        }
    }

    impl<M: ModIntBase> DpFactorial<M> {
        #[must_use]
        pub fn new() -> Self {
            Self {
                factorials: vec![M::new(1_u8)],
                inverse_factorials: vec![M::new(1_u8)],
                modulus: M::modulus(),
            }
        }

        fn ensure(&mut self, n: usize) {
            assert_eq!(
                M::modulus(),
                self.modulus,
                "the modulus changed after DpFactorial construction"
            );
            assert!(
                n < self.modulus as usize,
                "factorial inverse is not supported at or above the modulus"
            );
            if n < self.factorials.len() {
                return;
            }
            let old_len = self.factorials.len();
            self.factorials.reserve(n + 1 - old_len);
            self.inverse_factorials.reserve(n + 1 - old_len);
            for i in old_len..=n {
                let inverse = if i == 1 {
                    M::new(1_u8)
                } else {
                    let remainder = self.modulus as usize % i;
                    assert_ne!(remainder, 0, "factorial inverse does not exist");
                    -M::new(self.modulus as usize / i)
                        * self.factorials[remainder - 1]
                        * self.inverse_factorials[remainder]
                };
                let factorial = self.factorials[i - 1] * M::new(i);
                let inverse_factorial = self.inverse_factorials[i - 1] * inverse;
                self.factorials.push(factorial);
                self.inverse_factorials.push(inverse_factorial);
            }
        }

        pub fn factorial(&mut self, n: usize) -> M {
            self.ensure(n);
            self.factorials[n]
        }
        pub fn factorial_inverse(&mut self, n: usize) -> M {
            self.ensure(n);
            self.inverse_factorials[n]
        }
        pub fn combination(&mut self, n: usize, r: usize) -> M {
            if r > n {
                return M::new(0_u8);
            }
            self.ensure(n);
            self.factorials[n] * self.inverse_factorials[r] * self.inverse_factorials[n - r]
        }
        pub fn combination_small(&mut self, n: usize, r: usize) -> M {
            if r > n {
                return M::new(0_u8);
            }
            let r = r.min(n - r);
            self.ensure(r);
            if r == 0 {
                return M::new(1_u8);
            }
            ((n - r + 1)..=n).fold(self.inverse_factorials[r], |acc, x| acc * M::new(x))
        }
        pub fn permutation(&mut self, n: usize, r: usize) -> M {
            if r > n {
                return M::new(0_u8);
            }
            self.ensure(n);
            self.factorials[n] * self.inverse_factorials[n - r]
        }
    }
}
pub mod dsu {

    #[derive(Clone, Debug)]
    pub struct Dsu {
        n: usize,
        parent_or_size: Vec<i32>,
    }

    impl Dsu {
        pub fn new(size: usize) -> Self {
            Self {
                n: size,
                parent_or_size: vec![-1; size],
            }
        }

        pub fn merge(&mut self, a: usize, b: usize) -> usize {
            assert!(a < self.n);
            assert!(b < self.n);
            let (mut x, mut y) = (self.leader(a), self.leader(b));
            if x == y {
                return x;
            }
            if -self.parent_or_size[x] < -self.parent_or_size[y] {
                std::mem::swap(&mut x, &mut y);
            }
            self.parent_or_size[x] += self.parent_or_size[y];
            self.parent_or_size[y] = x as i32;
            x
        }

        pub fn same(&mut self, a: usize, b: usize) -> bool {
            assert!(a < self.n);
            assert!(b < self.n);
            self.leader(a) == self.leader(b)
        }

        pub fn leader(&mut self, a: usize) -> usize {
            assert!(a < self.n);
            if self.parent_or_size[a] < 0 {
                return a;
            }
            self.parent_or_size[a] = self.leader(self.parent_or_size[a] as usize) as i32;
            self.parent_or_size[a] as usize
        }

        pub fn size(&mut self, a: usize) -> usize {
            assert!(a < self.n);
            let x = self.leader(a);
            -self.parent_or_size[x] as usize
        }

        pub fn groups(&mut self) -> Vec<Vec<usize>> {
            let mut leader_buf = vec![0; self.n];
            let mut group_size = vec![0; self.n];
            for i in 0..self.n {
                leader_buf[i] = self.leader(i);
                group_size[leader_buf[i]] += 1;
            }
            let mut result = vec![Vec::new(); self.n];
            for i in 0..self.n {
                result[i].reserve(group_size[i]);
            }
            for i in 0..self.n {
                result[leader_buf[i]].push(i);
            }
            result
                .into_iter()
                .filter(|x| !x.is_empty())
                .collect::<Vec<Vec<usize>>>()
        }
    }
}
pub mod dsu_potential {
    #![warn(missing_docs)]

    #[derive(Copy, Clone, Debug, Eq, PartialEq)]
    pub enum PotentialMerge {
        Merged(usize),
        AlreadyConsistent(usize),
        Contradiction(usize),
    }

    #[derive(Clone, Debug)]
    pub struct DsuPotential {
        parent_or_size: Vec<isize>,
        weight_to_parent: Vec<i64>,
    }
    impl DsuPotential {
        #[must_use]
        pub fn new(size: usize) -> Self {
            Self {
                parent_or_size: vec![-1; size],
                weight_to_parent: vec![0; size],
            }
        }
        #[must_use]
        pub fn len(&self) -> usize {
            self.parent_or_size.len()
        }
        #[must_use]
        pub fn is_empty(&self) -> bool {
            self.len() == 0
        }
        pub fn merge(&mut self, a: usize, b: usize, difference: i64) -> PotentialMerge {
            self.check(a);
            self.check(b);
            let pa = self.potential(a);
            let pb = self.potential(b);
            let (mut x, mut y) = (self.leader(a), self.leader(b));
            let mut root_difference = difference
                .checked_add(pa)
                .and_then(|v| v.checked_sub(pb))
                .expect("potential overflow");
            if x == y {
                return if root_difference == 0 {
                    PotentialMerge::AlreadyConsistent(x)
                } else {
                    PotentialMerge::Contradiction(x)
                };
            }
            if -self.parent_or_size[x] < -self.parent_or_size[y] {
                std::mem::swap(&mut x, &mut y);
                root_difference = root_difference.checked_neg().expect("potential overflow");
            }
            self.parent_or_size[x] += self.parent_or_size[y];
            self.parent_or_size[y] = x as isize;
            self.weight_to_parent[y] = root_difference;
            PotentialMerge::Merged(x)
        }
        pub fn same(&mut self, a: usize, b: usize) -> bool {
            self.check(a);
            self.check(b);
            self.leader(a) == self.leader(b)
        }
        pub fn leader(&mut self, vertex: usize) -> usize {
            self.check(vertex);
            let mut root = vertex;
            let mut path = Vec::new();
            while self.parent_or_size[root] >= 0 {
                path.push(root);
                root = self.parent_or_size[root] as usize;
            }
            let mut total = 0_i64;
            for node in path.into_iter().rev() {
                total = total
                    .checked_add(self.weight_to_parent[node])
                    .expect("potential overflow");
                self.parent_or_size[node] = root as isize;
                self.weight_to_parent[node] = total;
            }
            root
        }
        pub fn potential(&mut self, vertex: usize) -> i64 {
            self.leader(vertex);
            self.weight_to_parent[vertex]
        }
        pub fn diff(&mut self, a: usize, b: usize) -> Option<i64> {
            if !self.same(a, b) {
                return None;
            }
            self.potential(b).checked_sub(self.potential(a))
        }
        pub fn size(&mut self, vertex: usize) -> usize {
            let root = self.leader(vertex);
            (-self.parent_or_size[root]) as usize
        }
        pub fn groups(&mut self) -> Vec<Vec<usize>> {
            let mut group_indices = vec![usize::MAX; self.len()];
            let mut groups = Vec::<Vec<usize>>::new();
            for v in 0..self.len() {
                let root = self.leader(v);
                let index = if group_indices[root] == usize::MAX {
                    group_indices[root] = groups.len();
                    groups.push(Vec::new());
                    groups.len() - 1
                } else {
                    group_indices[root]
                };
                groups[index].push(v);
            }
            groups
        }
        fn check(&self, v: usize) {
            assert!(v < self.len(), "vertex index out of bounds");
        }
    }
}
pub mod eratosthenes {
    #![warn(missing_docs)]

    use std::convert::TryFrom;
    use std::error::Error;
    use std::fmt;

    #[derive(Clone, Debug)]
    pub struct EratosthenesSieve {
        smallest: Vec<usize>,
        primes: Vec<usize>,
    }
    impl EratosthenesSieve {
        #[must_use]
        pub fn new(limit: usize) -> Self {
            let mut smallest = vec![0; limit.checked_add(1).expect("sieve limit overflow")];
            let mut primes = Vec::new();
            for value in 2..=limit {
                if smallest[value] == 0 {
                    smallest[value] = value;
                    primes.push(value);
                }
                for &prime in &primes {
                    if prime > smallest[value] || prime > limit / value {
                        break;
                    }
                    smallest[value * prime] = prime;
                }
            }
            Self { smallest, primes }
        }
        #[must_use]
        pub fn get_factor(&self, value: usize) -> Option<usize> {
            assert!(value < self.smallest.len(), "value exceeds sieve limit");
            (value >= 2).then(|| self.smallest[value])
        }
        #[must_use]
        pub fn is_prime(&self, value: usize) -> bool {
            self.get_factor(value) == Some(value)
        }
        pub fn prime_factorization(&self, mut value: usize) -> std::vec::IntoIter<usize> {
            assert!(value < self.smallest.len(), "value exceeds sieve limit");
            let mut factors = Vec::new();
            while value >= 2 {
                let p = self.smallest[value];
                factors.push(p);
                value /= p;
            }
            factors.into_iter()
        }
        #[must_use]
        pub fn primes(&self) -> &[usize] {
            &self.primes
        }
    }

    #[derive(Clone, Debug, Eq, PartialEq)]
    pub enum SieveRangeError {
        ReversedRange,
        RangeTooLarge,
    }
    impl fmt::Display for SieveRangeError {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "{self:?}")
        }
    }
    impl Error for SieveRangeError {}

    #[derive(Clone, Debug)]
    pub struct EratosthenesSieveRange {
        start: u64,
        end: u64,
        is_prime: Vec<bool>,
        base_primes: Vec<u64>,
    }
    impl EratosthenesSieveRange {
        pub fn new(start: u64, end: u64) -> Result<Self, SieveRangeError> {
            if start > end {
                return Err(SieveRangeError::ReversedRange);
            }
            let width = end
                .checked_sub(start)
                .and_then(|x| x.checked_add(1))
                .and_then(|x| usize::try_from(x).ok())
                .ok_or(SieveRangeError::RangeTooLarge)?;
            let root = end.isqrt();
            let base = EratosthenesSieve::new(
                usize::try_from(root).map_err(|_| SieveRangeError::RangeTooLarge)?,
            );
            let base_primes = base.primes().iter().map(|&x| x as u64).collect::<Vec<_>>();
            let mut is_prime = vec![true; width];
            for &p in &base_primes {
                let square = p * p;
                let first = start.div_ceil(p).saturating_mul(p).max(square);
                if first > end {
                    continue;
                }
                let mut value = first;
                loop {
                    is_prime[(value - start) as usize] = false;
                    match value.checked_add(p) {
                        Some(next) if next <= end => value = next,
                        _ => break,
                    }
                }
            }
            if start == 0 {
                is_prime[0] = false;
                if end >= 1 {
                    is_prime[1] = false;
                }
            } else if start == 1 {
                is_prime[0] = false;
            }
            Ok(Self {
                start,
                end,
                is_prime,
                base_primes,
            })
        }
        #[must_use]
        pub fn is_prime(&self, value: u64) -> Option<bool> {
            (self.start..=self.end)
                .contains(&value)
                .then(|| self.is_prime[(value - self.start) as usize])
        }
        pub fn prime_factorization(&self, value: u64) -> Option<Vec<u64>> {
            if !(self.start..=self.end).contains(&value) {
                return None;
            }
            let mut value = value;
            let mut factors = Vec::new();
            for &p in &self.base_primes {
                if p > value / p {
                    break;
                }
                while value.is_multiple_of(p) {
                    factors.push(p);
                    value /= p;
                }
            }
            if value > 1 {
                factors.push(value);
            }
            Some(factors)
        }
    }
}
pub mod fenwicktree {

    use std::ops::{Bound, RangeBounds};

    #[derive(Clone)]
    pub struct FenwickTree<T> {
        n: usize,
        ary: Vec<T>,
        e: T,
    }

    impl<T: Clone + std::ops::AddAssign<T>> FenwickTree<T> {
        pub fn new(n: usize, e: T) -> Self {
            FenwickTree {
                n,
                ary: vec![e.clone(); n],
                e,
            }
        }

        pub fn accum(&self, mut idx: usize) -> T {
            let mut sum = self.e.clone();
            while idx > 0 {
                sum += self.ary[idx - 1].clone();
                idx &= idx - 1;
            }
            sum
        }

        pub fn add<U: Clone>(&mut self, mut idx: usize, val: U)
        where
            T: std::ops::AddAssign<U>,
        {
            let n = self.n;
            idx += 1;
            while idx <= n {
                self.ary[idx - 1] += val.clone();
                idx += idx & idx.wrapping_neg();
            }
        }

        pub fn sum<R>(&self, range: R) -> T
        where
            T: std::ops::Sub<Output = T>,
            R: RangeBounds<usize>,
        {
            let r = match range.end_bound() {
                Bound::Included(r) => r + 1,
                Bound::Excluded(r) => *r,
                Bound::Unbounded => self.n,
            };
            let l = match range.start_bound() {
                Bound::Included(l) => *l,
                Bound::Excluded(l) => l + 1,
                Bound::Unbounded => return self.accum(r),
            };
            self.accum(r) - self.accum(l)
        }
    }
}
pub mod fft {
    #![warn(missing_docs)]

    use crate::Complex;
    use std::error::Error;
    use std::f64::consts::TAU;
    use std::fmt;

    #[derive(Copy, Clone, Debug, Eq, PartialEq)]
    pub enum FftDirection {
        Forward,
        Inverse,
    }

    #[derive(Copy, Clone, Debug, Eq, PartialEq)]
    pub enum FftError {
        Empty,
        NonPowerOfTwo(usize),
    }

    impl fmt::Display for FftError {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            match self {
                Self::Empty => write!(f, "FFT input must not be empty"),
                Self::NonPowerOfTwo(n) => write!(f, "FFT length {n} is not a power of two"),
            }
        }
    }
    impl Error for FftError {}

    pub fn fft(values: &mut [Complex<f64>], direction: FftDirection) -> Result<(), FftError> {
        let n = values.len();
        if n == 0 {
            return Err(FftError::Empty);
        }
        if !n.is_power_of_two() {
            return Err(FftError::NonPowerOfTwo(n));
        }
        if n == 1 {
            return Ok(());
        }
        let shift = usize::BITS - n.trailing_zeros();
        for i in 0..n {
            let j = i.reverse_bits() >> shift;
            if i < j {
                values.swap(i, j);
            }
        }
        let sign = if direction == FftDirection::Forward {
            -1.0
        } else {
            1.0
        };
        let mut width = 2;
        while width <= n {
            let (sin, cos) = (sign * TAU / width as f64).sin_cos();
            let root = Complex::new(cos, sin);
            for chunk in values.chunks_exact_mut(width) {
                let mut factor = Complex::new(1.0, 0.0);
                let half = width / 2;
                for i in 0..half {
                    let even = chunk[i];
                    let odd = chunk[i + half] * factor;
                    chunk[i] = even + odd;
                    chunk[i + half] = even - odd;
                    factor *= root;
                }
            }
            width *= 2;
        }
        if direction == FftDirection::Inverse {
            for value in values {
                value.re /= n as f64;
                value.im /= n as f64;
            }
        }
        Ok(())
    }

    pub fn fft_padded(
        values: &[Complex<f64>],
        direction: FftDirection,
    ) -> Result<Vec<Complex<f64>>, FftError> {
        if values.is_empty() {
            return Ok(Vec::new());
        }
        let mut output = values.to_vec();
        output.resize(values.len().next_power_of_two(), Complex::new(0.0, 0.0));
        fft(&mut output, direction)?;
        Ok(output)
    }
}
pub mod geometry {
    #![warn(missing_docs)]

    use std::cmp::Ordering;
    use std::ops::{Add, AddAssign, Mul, Neg, Sub, SubAssign};

    #[derive(Debug, Copy, Clone, PartialEq, Eq, Default)]
    pub struct Point<T> {
        pub x: T,
        pub y: T,
    }
    impl<T> Point<T> {
        #[must_use]
        pub const fn new(x: T, y: T) -> Self {
            Self { x, y }
        }
    }
    impl<T: Add<Output = T>> Add for Point<T> {
        type Output = Self;
        fn add(self, rhs: Self) -> Self {
            Self::new(self.x + rhs.x, self.y + rhs.y)
        }
    }
    impl<T: AddAssign> AddAssign for Point<T> {
        fn add_assign(&mut self, rhs: Self) {
            self.x += rhs.x;
            self.y += rhs.y;
        }
    }
    impl<T: Sub<Output = T>> Sub for Point<T> {
        type Output = Self;
        fn sub(self, rhs: Self) -> Self {
            Self::new(self.x - rhs.x, self.y - rhs.y)
        }
    }
    impl<T: SubAssign> SubAssign for Point<T> {
        fn sub_assign(&mut self, rhs: Self) {
            self.x -= rhs.x;
            self.y -= rhs.y;
        }
    }

    #[must_use]
    pub fn cross<T>(a: Point<T>, b: Point<T>) -> T
    where
        T: Mul<Output = T> + Sub<Output = T>,
    {
        a.x * b.y - a.y * b.x
    }
    #[must_use]
    pub fn dot<T>(a: Point<T>, b: Point<T>) -> T
    where
        T: Mul<Output = T> + Add<Output = T>,
    {
        a.x * b.x + a.y * b.y
    }
    #[must_use]
    pub fn norm<T>(a: Point<T>) -> T
    where
        T: Copy + Mul<Output = T> + Add<Output = T>,
    {
        dot(a, a)
    }

    #[derive(Copy, Clone, Debug, Eq, PartialEq)]
    pub enum Ccw {
        CounterClockwise,
        Clockwise,
        OnlineBack,
        OnlineFront,
        OnSegment,
    }
    #[must_use]
    pub fn ccw<T>(a: Point<T>, b: Point<T>, c: Point<T>) -> Ccw
    where
        T: Copy + Default + PartialOrd + Sub<Output = T> + Mul<Output = T> + Add<Output = T>,
    {
        let b = b - a;
        let c = c - a;
        let zero = T::default();
        let area = cross(b, c);
        if area > zero {
            Ccw::CounterClockwise
        } else if area < zero {
            Ccw::Clockwise
        } else if dot(b, c) < zero {
            Ccw::OnlineBack
        } else if norm(b) < norm(c) {
            Ccw::OnlineFront
        } else {
            Ccw::OnSegment
        }
    }

    fn upper_half<T>(p: &Point<T>) -> bool
    where
        T: Default + PartialOrd,
    {
        p.y > T::default() || p.y == T::default() && p.x >= T::default()
    }
    pub fn polar_cmp<T>(a: &Point<T>, b: &Point<T>) -> Ordering
    where
        T: Copy + Default + Ord + Mul<Output = T> + Sub<Output = T> + Add<Output = T>,
    {
        match upper_half(a).cmp(&upper_half(b)).reverse() {
            Ordering::Equal => match cross(*a, *b).cmp(&T::default()).reverse() {
                Ordering::Equal => norm(*a).cmp(&norm(*b)),
                order => order,
            },
            order => order,
        }
    }

    #[derive(Copy, Clone, Debug, PartialEq, Eq)]
    pub struct Line<T> {
        a: T,
        b: T,
        c: T,
    }
    impl<T> Line<T> {
        pub fn coefficients(&self) -> (&T, &T, &T) {
            (&self.a, &self.b, &self.c)
        }
    }
    impl<T> Line<T>
    where
        T: Copy + Add<Output = T> + Sub<Output = T> + Mul<Output = T> + Neg<Output = T>,
    {
        pub fn from_points(p: Point<T>, q: Point<T>) -> Option<Self>
        where
            T: PartialEq,
        {
            if p == q {
                return None;
            }
            let a = p.y - q.y;
            let b = q.x - p.x;
            let c = -(a * p.x + b * p.y);
            Some(Self { a, b, c })
        }
        pub fn is_parallel(&self, other: &Self) -> bool
        where
            T: PartialEq,
        {
            self.a * other.b == other.a * self.b
        }
        pub fn is_orthogonal(&self, other: &Self) -> bool
        where
            T: Add<Output = T> + Default + PartialEq,
        {
            self.a * other.a + self.b * other.b == T::default()
        }
    }

    #[derive(Copy, Clone, Debug, PartialEq)]
    pub enum LineIntersection {
        Parallel,
        Coincident,
        Point(Point<f64>),
    }
    impl Line<f64> {
        pub fn intersection(&self, other: &Self, epsilon: f64) -> LineIntersection {
            assert!(epsilon >= 0.0);
            let determinant = self.a * other.b - other.a * self.b;
            if determinant.abs() <= epsilon {
                let ac = self.a * other.c - other.a * self.c;
                let bc = self.b * other.c - other.b * self.c;
                return if ac.abs() <= epsilon && bc.abs() <= epsilon {
                    LineIntersection::Coincident
                } else {
                    LineIntersection::Parallel
                };
            }
            LineIntersection::Point(Point::new(
                (self.b * other.c - other.b * self.c) / determinant,
                (self.c * other.a - other.c * self.a) / determinant,
            ))
        }
    }

    pub fn convex_hull<T>(points: &[Point<T>]) -> Vec<Point<T>>
    where
        T: Copy + Ord + Sub<Output = T> + Mul<Output = T> + Default,
    {
        let mut points = points.to_vec();
        points.sort_unstable_by(|a, b| a.x.cmp(&b.x).then_with(|| a.y.cmp(&b.y)));
        points.dedup();
        if points.len() <= 1 {
            return points;
        }
        let turn = |a: Point<T>, b: Point<T>, c: Point<T>| cross(b - a, c - a);
        let mut lower = Vec::new();
        for &p in &points {
            while lower.len() >= 2
                && turn(lower[lower.len() - 2], lower[lower.len() - 1], p) <= T::default()
            {
                lower.pop();
            }
            lower.push(p);
        }
        let mut upper = Vec::new();
        for &p in points.iter().rev() {
            while upper.len() >= 2
                && turn(upper[upper.len() - 2], upper[upper.len() - 1], p) <= T::default()
            {
                upper.pop();
            }
            upper.push(p);
        }
        lower.pop();
        upper.pop();
        lower.extend(upper);
        lower
    }
}
pub mod global {
    #![warn(missing_docs)]

    #[macro_export]
    macro_rules! global_init {
    ($name:ident : $ty:ty = $initial:expr $(,)?) => {
        std::thread_local! { pub static $name: std::cell::RefCell<$ty> = std::cell::RefCell::new($initial); }
    };
}

    #[macro_export]
    macro_rules! global_with {
        ($name:ident, |$value:ident| $body:expr $(,)?) => {
            $name.with_borrow(|$value| $body)
        };
    }

    #[macro_export]
    macro_rules! global_with_mut {
        ($name:ident, |$value:ident| $body:expr $(,)?) => {
            $name.with_borrow_mut(|$value| $body)
        };
    }
}
pub mod graph {
    #![warn(missing_docs)]

    use std::collections::{BTreeSet, VecDeque};
    use std::error::Error;
    use std::fmt;

    #[derive(Clone, Debug, Eq, PartialEq)]
    pub enum GraphError {
        InvalidVertex(usize),
        NegativeEdge { from: usize, to: usize, weight: i64 },
        DistanceOverflow,
        NonSquareMatrix,
    }
    impl fmt::Display for GraphError {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "{self:?}")
        }
    }
    impl Error for GraphError {}

    #[derive(Clone, Debug, Eq, PartialEq)]
    pub struct BellmanFordResult {
        pub distances: Vec<Option<i64>>,
        pub affected_by_negative_cycle: Vec<bool>,
    }

    fn validate(graph: &[Vec<(usize, i64)>], start: usize) -> Result<(), GraphError> {
        if start >= graph.len() {
            return Err(GraphError::InvalidVertex(start));
        }
        for edges in graph {
            for &(to, _) in edges {
                if to >= graph.len() {
                    return Err(GraphError::InvalidVertex(to));
                }
            }
        }
        Ok(())
    }

    pub fn bellman_ford(
        graph: &[Vec<(usize, i64)>],
        start: usize,
    ) -> Result<BellmanFordResult, GraphError> {
        validate(graph, start)?;
        let n = graph.len();
        let sources = graph
            .iter()
            .enumerate()
            .filter(|(_, edges)| !edges.is_empty())
            .collect::<Vec<_>>();
        let mut distances: Vec<Option<i64>> = vec![None; n];
        distances[start] = Some(0);
        for _ in 1..n {
            let mut changed = false;
            for &(from, edges) in &sources {
                let Some(base) = distances[from] else {
                    continue;
                };
                for &(to, weight) in edges {
                    let candidate = base
                        .checked_add(weight)
                        .ok_or(GraphError::DistanceOverflow)?;
                    if distances[to].is_none_or(|old| candidate < old) {
                        distances[to] = Some(candidate);
                        changed = true;
                    }
                }
            }
            if !changed {
                break;
            }
        }
        let mut affected = vec![false; n];
        let mut queue = VecDeque::new();
        for &(from, edges) in &sources {
            let Some(base) = distances[from] else {
                continue;
            };
            for &(to, weight) in edges {
                let candidate = base
                    .checked_add(weight)
                    .ok_or(GraphError::DistanceOverflow)?;
                if distances[to].is_some_and(|old| candidate < old) && !affected[to] {
                    affected[to] = true;
                    queue.push_back(to);
                }
            }
        }
        while let Some(from) = queue.pop_front() {
            for &(to, _) in &graph[from] {
                if !affected[to] {
                    affected[to] = true;
                    queue.push_back(to);
                }
            }
        }
        Ok(BellmanFordResult {
            distances,
            affected_by_negative_cycle: affected,
        })
    }

    pub fn dijkstra(
        graph: &[Vec<(usize, i64)>],
        start: usize,
    ) -> Result<Vec<Option<i64>>, GraphError> {
        validate(graph, start)?;
        for (from, edges) in graph.iter().enumerate() {
            for &(to, weight) in edges {
                if weight < 0 {
                    return Err(GraphError::NegativeEdge { from, to, weight });
                }
            }
        }
        let mut distances: Vec<Option<i64>> = vec![None; graph.len()];
        let mut queue = BTreeSet::new();
        distances[start] = Some(0);
        queue.insert((0_i64, start));
        while let Some((distance, from)) = queue.pop_first() {
            for &(to, weight) in &graph[from] {
                let candidate = distance
                    .checked_add(weight)
                    .ok_or(GraphError::DistanceOverflow)?;
                if distances[to].is_none_or(|old| candidate < old) {
                    if let Some(old) = distances[to] {
                        queue.remove(&(old, to));
                    }
                    distances[to] = Some(candidate);
                    queue.insert((candidate, to));
                }
            }
        }
        Ok(distances)
    }

    pub fn warshall_floyd(matrix: &mut [Vec<Option<i64>>]) -> Result<(), GraphError> {
        let n = matrix.len();
        if matrix.iter().any(|row| row.len() != n) {
            return Err(GraphError::NonSquareMatrix);
        }
        for k in 0..n {
            for i in 0..n {
                for j in 0..n {
                    if let (Some(a), Some(b)) = (matrix[i][k], matrix[k][j]) {
                        let candidate = a.checked_add(b).ok_or(GraphError::DistanceOverflow)?;
                        if matrix[i][j].is_none_or(|old| candidate < old) {
                            matrix[i][j] = Some(candidate);
                        }
                    }
                }
            }
        }
        Ok(())
    }
}
pub mod graph_search {
    #![warn(missing_docs)]

    use std::collections::VecDeque;
    use std::error::Error;
    use std::fmt;

    #[derive(Debug, PartialEq, Eq, Clone, Copy)]
    pub enum DfsOrder {
        Preorder,
        Postorder,
    }

    #[derive(Clone, Debug, Eq, PartialEq)]
    pub enum GraphSearchError {
        InvalidVertex(usize),
        CycleDetected,
    }
    impl fmt::Display for GraphSearchError {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "{self:?}")
        }
    }
    impl Error for GraphSearchError {}

    fn validate(graph: &[Vec<usize>]) -> Result<(), GraphSearchError> {
        for edges in graph {
            for &to in edges {
                if to >= graph.len() {
                    return Err(GraphSearchError::InvalidVertex(to));
                }
            }
        }
        Ok(())
    }

    pub fn topological_sort(graph: &[Vec<usize>]) -> Result<Vec<usize>, GraphSearchError> {
        validate(graph)?;
        let mut indegree = vec![0; graph.len()];
        for edges in graph {
            for &to in edges {
                indegree[to] += 1;
            }
        }
        let mut queue = (0..graph.len())
            .filter(|&v| indegree[v] == 0)
            .collect::<VecDeque<_>>();
        let mut order = Vec::with_capacity(graph.len());
        while let Some(from) = queue.pop_front() {
            order.push(from);
            for &to in &graph[from] {
                indegree[to] -= 1;
                if indegree[to] == 0 {
                    queue.push_back(to);
                }
            }
        }
        if order.len() == graph.len() {
            Ok(order)
        } else {
            Err(GraphSearchError::CycleDetected)
        }
    }

    pub fn graph_dfs(
        start: usize,
        graph: &[Vec<usize>],
        order: DfsOrder,
    ) -> Result<Vec<usize>, GraphSearchError> {
        validate(graph)?;
        if start >= graph.len() {
            return Err(GraphSearchError::InvalidVertex(start));
        }
        let mut visited = vec![false; graph.len()];
        let mut output = Vec::new();
        let mut stack = vec![(start, 0)];
        visited[start] = true;
        if order == DfsOrder::Preorder {
            output.push(start);
        }
        while let Some((vertex, next_edge)) = stack.last_mut() {
            if *next_edge == graph[*vertex].len() {
                let (vertex, _) = stack.pop().unwrap();
                if order == DfsOrder::Postorder {
                    output.push(vertex);
                }
                continue;
            }
            let next = graph[*vertex][*next_edge];
            *next_edge += 1;
            if !visited[next] {
                visited[next] = true;
                if order == DfsOrder::Preorder {
                    output.push(next);
                }
                stack.push((next, 0));
            }
        }
        Ok(output)
    }

    pub fn graph_bfs(start: usize, graph: &[Vec<usize>]) -> Result<Vec<usize>, GraphSearchError> {
        validate(graph)?;
        if start >= graph.len() {
            return Err(GraphSearchError::InvalidVertex(start));
        }
        let mut visited = vec![false; graph.len()];
        let mut queue = VecDeque::from([start]);
        let mut output = Vec::new();
        visited[start] = true;
        while let Some(from) = queue.pop_front() {
            output.push(from);
            for &to in &graph[from] {
                if !visited[to] {
                    visited[to] = true;
                    queue.push_back(to);
                }
            }
        }
        Ok(output)
    }
}
pub mod input {
    #![warn(missing_docs)]

    use std::error::Error;
    use std::fmt;
    use std::io::{self, Read};
    use std::str::FromStr;

    #[derive(Debug)]
    pub enum ScanError {
        Io(io::Error),
        UnexpectedEof,
        Parse {
            index: usize,
            token: String,
            message: String,
        },
    }
    impl fmt::Display for ScanError {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            match self {
                Self::Io(e) => write!(f, "input I/O error: {e}"),
                Self::UnexpectedEof => write!(f, "unexpected end of input"),
                Self::Parse {
                    index,
                    token,
                    message,
                } => write!(f, "cannot parse token {index} ({token:?}): {message}"),
            }
        }
    }
    impl Error for ScanError {
        fn source(&self) -> Option<&(dyn Error + 'static)> {
            match self {
                Self::Io(e) => Some(e),
                _ => None,
            }
        }
    }
    impl From<io::Error> for ScanError {
        fn from(value: io::Error) -> Self {
            Self::Io(value)
        }
    }

    #[derive(Clone, Debug)]
    pub struct Scanner {
        tokens: Vec<String>,
        index: usize,
    }
    impl Scanner {
        pub fn new(mut reader: impl Read) -> Result<Self, ScanError> {
            let mut source = String::new();
            reader.read_to_string(&mut source)?;
            Ok(Self {
                tokens: source.split_whitespace().map(str::to_owned).collect(),
                index: 0,
            })
        }
        pub fn stdin() -> Result<Self, ScanError> {
            Self::new(io::stdin())
        }
        pub fn next_token<T>(&mut self) -> Result<T, ScanError>
        where
            T: FromStr,
            T::Err: fmt::Display,
        {
            let token = self
                .tokens
                .get(self.index)
                .ok_or(ScanError::UnexpectedEof)?
                .clone();
            let index = self.index;
            self.index += 1;
            token.parse().map_err(|e: T::Err| ScanError::Parse {
                index,
                token,
                message: e.to_string(),
            })
        }
        pub fn next_vec<T>(&mut self, len: usize) -> Result<Vec<T>, ScanError>
        where
            T: FromStr,
            T::Err: fmt::Display,
        {
            (0..len).map(|_| self.next_token()).collect()
        }
        #[must_use]
        pub fn remaining(&self) -> usize {
            self.tokens.len() - self.index
        }
    }

    #[macro_export]
    macro_rules! inputv{($scanner:expr;)=>{};($scanner:expr;mut $name:ident:$ty:ty $(,$($rest:tt)*)?)=>{let mut $name:$ty=$scanner.next_token().expect("failed to parse input");$crate::inputv!($scanner;$($($rest)*)?);};($scanner:expr;$name:ident:$ty:ty $(,$($rest:tt)*)?)=>{let $name:$ty=$scanner.next_token().expect("failed to parse input");$crate::inputv!($scanner;$($($rest)*)?);};}

    pub trait ToCharVec {
        fn to_charvec(&self) -> Vec<char>;
    }
    impl ToCharVec for str {
        fn to_charvec(&self) -> Vec<char> {
            self.chars().collect()
        }
    }
    impl ToCharVec for String {
        fn to_charvec(&self) -> Vec<char> {
            self.as_str().to_charvec()
        }
    }
}
pub mod internal_bit {

    #[allow(dead_code)]
    pub(crate) fn ceil_pow2(n: u32) -> u32 {
        32 - n.saturating_sub(1).leading_zeros()
    }
}
pub mod internal_math {

    #![allow(dead_code)]
    use std::{mem::swap, num::Wrapping as W};

    /* const */
    pub(crate) fn safe_mod(mut x: i64, m: i64) -> i64 {
        x %= m;
        if x < 0 {
            x += m;
        }
        x
    }

    pub(crate) struct Barrett {
        pub(crate) _m: u32,
        pub(crate) im: u64,
    }

    impl Barrett {
        pub(crate) fn new(m: u32) -> Barrett {
            Barrett {
                _m: m,
                im: (-1i64 as u64 / m as u64).wrapping_add(1),
            }
        }

        pub(crate) fn umod(&self) -> u32 {
            self._m
        }

        #[allow(clippy::many_single_char_names)]
        pub(crate) fn mul(&self, a: u32, b: u32) -> u32 {
            mul_mod(a, b, self._m, self.im)
        }
    }

    #[allow(clippy::many_single_char_names)]
    pub(crate) fn mul_mod(a: u32, b: u32, m: u32, im: u64) -> u32 {
        let mut z = a as u64;
        z *= b as u64;
        let x = (((z as u128) * (im as u128)) >> 64) as u64;
        let mut v = z.wrapping_sub(x.wrapping_mul(m as u64)) as u32;
        if m <= v {
            v = v.wrapping_add(m);
        }
        v
    }

    /* const */
    #[allow(clippy::many_single_char_names)]
    pub(crate) fn pow_mod(x: i64, mut n: i64, m: i32) -> i64 {
        if m == 1 {
            return 0;
        }
        let _m = m as u32;
        let mut r: u64 = 1;
        let mut y: u64 = safe_mod(x, m as i64) as u64;
        while n != 0 {
            if (n & 1) > 0 {
                r = (r * y) % (_m as u64);
            }
            y = (y * y) % (_m as u64);
            n >>= 1;
        }
        r as i64
    }

    /* const */
    pub(crate) fn is_prime(n: i32) -> bool {
        let n = n as i64;
        match n {
            _ if n <= 1 => return false,
            2 | 7 | 61 => return true,
            _ if n % 2 == 0 => return false,
            _ => {}
        }
        let mut d = n - 1;
        while d % 2 == 0 {
            d /= 2;
        }
        for &a in &[2, 7, 61] {
            let mut t = d;
            let mut y = pow_mod(a, t, n as i32);
            while t != n - 1 && y != 1 && y != n - 1 {
                y = y * y % n;
                t <<= 1;
            }
            if y != n - 1 && t % 2 == 0 {
                return false;
            }
        }
        true
    }

    /* const */
    #[allow(clippy::many_single_char_names)]
    pub(crate) fn inv_gcd(a: i64, b: i64) -> (i64, i64) {
        let a = safe_mod(a, b);
        if a == 0 {
            return (b, 0);
        }

        let mut s = b;
        let mut t = a;
        let mut m0 = 0;
        let mut m1 = 1;

        while t != 0 {
            let u = s / t;
            s -= t * u;
            m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b

            swap(&mut s, &mut t);
            swap(&mut m0, &mut m1);
        }
        if m0 < 0 {
            m0 += b / s;
        }
        (s, m0)
    }

    /* const */
    pub(crate) fn primitive_root(m: i32) -> i32 {
        match m {
            2 => return 1,
            167_772_161 => return 3,
            469_762_049 => return 3,
            754_974_721 => return 11,
            998_244_353 => return 3,
            _ => {}
        }

        let mut divs = [0; 20];
        divs[0] = 2;
        let mut cnt = 1;
        let mut x = (m - 1) / 2;
        while x % 2 == 0 {
            x /= 2;
        }
        for i in (3..i32::MAX).step_by(2) {
            if i as i64 * i as i64 > x as i64 {
                break;
            }
            if x % i == 0 {
                divs[cnt] = i;
                cnt += 1;
                while x % i == 0 {
                    x /= i;
                }
            }
        }
        if x > 1 {
            divs[cnt] = x;
            cnt += 1;
        }
        let mut g = 2;
        loop {
            if (0..cnt).all(|i| pow_mod(g, ((m - 1) / divs[i]) as i64, m) != 1) {
                break g as i32;
            }
            g += 1;
        }
    }

    /* const */
    #[allow(clippy::many_single_char_names)]
    pub(crate) fn floor_sum_unsigned(
        mut n: W<u64>,
        mut m: W<u64>,
        mut a: W<u64>,
        mut b: W<u64>,
    ) -> W<u64> {
        let mut ans = W(0);
        loop {
            if a >= m {
                if n > W(0) {
                    ans += n * (n - W(1)) / W(2) * (a / m);
                }
                a %= m;
            }
            if b >= m {
                ans += n * (b / m);
                b %= m;
            }

            let y_max = a * n + b;
            if y_max < m {
                break;
            }
            n = y_max / m;
            b = y_max % m;
            std::mem::swap(&mut m, &mut a);
        }
        ans
    }
}
pub mod internal_queue {

    #![allow(dead_code)]

    #[derive(Default)]
    pub(crate) struct SimpleQueue<T> {
        payload: Vec<T>,
        pos: usize,
    }

    impl<T> SimpleQueue<T> {
        pub(crate) fn reserve(&mut self, n: usize) {
            if n > self.payload.len() {
                self.payload.reserve(n - self.payload.len());
            }
        }

        pub(crate) fn size(&self) -> usize {
            self.payload.len() - self.pos
        }

        pub(crate) fn empty(&self) -> bool {
            self.pos == self.payload.len()
        }

        pub(crate) fn push(&mut self, t: T) {
            self.payload.push(t);
        }

        pub(crate) fn front(&self) -> Option<&T> {
            if self.pos < self.payload.len() {
                Some(&self.payload[self.pos])
            } else {
                None
            }
        }

        pub(crate) fn clear(&mut self) {
            self.payload.clear();
            self.pos = 0;
        }

        pub(crate) fn pop(&mut self) -> Option<&T> {
            if self.pos < self.payload.len() {
                self.pos += 1;
                Some(&self.payload[self.pos - 1])
            } else {
                None
            }
        }
    }
}
pub mod internal_scc {

    pub struct Csr<E> {
        start: Vec<usize>,
        elist: Vec<E>,
    }

    impl<E> Csr<E>
    where
        E: Copy,
    {
        pub fn new(n: usize, edges: &[(usize, E)], init: E) -> Self {
            let mut csr = Csr {
                start: vec![0; n + 1],
                elist: vec![init; edges.len()],
            };
            for e in edges.iter() {
                csr.start[e.0 + 1] += 1;
            }
            for i in 1..=n {
                csr.start[i] += csr.start[i - 1];
            }
            let mut counter = csr.start.clone();
            for e in edges.iter() {
                csr.elist[counter[e.0]] = e.1;
                counter[e.0] += 1;
            }
            csr
        }
    }

    #[derive(Copy, Clone, Debug)]
    struct _Edge {
        to: usize,
    }

    #[derive(Clone, Debug)]
    pub struct SccGraph {
        n: usize,
        edges: Vec<(usize, _Edge)>,
    }

    impl SccGraph {
        pub fn new(n: usize) -> Self {
            SccGraph { n, edges: vec![] }
        }

        pub fn num_vertices(&self) -> usize {
            self.n
        }

        pub fn add_edge(&mut self, from: usize, to: usize) {
            self.edges.push((from, _Edge { to }));
        }

        pub fn scc_ids(&self) -> (usize, Vec<usize>) {
            struct _Env {
                g: Csr<_Edge>,
                now_ord: usize,
                group_num: usize,
                visited: Vec<usize>,
                low: Vec<usize>,
                ord: Vec<Option<usize>>,
                ids: Vec<usize>,
            }
            let mut env = _Env {
                g: Csr::new(self.n, &self.edges, _Edge { to: 0 }),
                now_ord: 0,
                group_num: 0,
                visited: Vec::with_capacity(self.n),
                low: vec![0; self.n],
                ord: vec![None; self.n],
                ids: vec![0; self.n],
            };

            fn dfs(v: usize, n: usize, env: &mut _Env) {
                env.low[v] = env.now_ord;
                env.ord[v] = Some(env.now_ord);
                env.now_ord += 1;
                env.visited.push(v);

                for i in env.g.start[v]..env.g.start[v + 1] {
                    let to = env.g.elist[i].to;
                    if let Some(x) = env.ord[to] {
                        env.low[v] = std::cmp::min(env.low[v], x);
                    } else {
                        dfs(to, n, env);
                        env.low[v] = std::cmp::min(env.low[v], env.low[to]);
                    }
                }
                if env.low[v] == env.ord[v].unwrap() {
                    loop {
                        let u = *env.visited.last().unwrap();
                        env.visited.pop();
                        env.ord[u] = Some(n);
                        env.ids[u] = env.group_num;
                        if u == v {
                            break;
                        }
                    }
                    env.group_num += 1;
                }
            }
            for i in 0..self.n {
                if env.ord[i].is_none() {
                    dfs(i, self.n, &mut env);
                }
            }
            for x in env.ids.iter_mut() {
                *x = env.group_num - 1 - *x;
            }
            (env.group_num, env.ids)
        }

        #[allow(clippy::redundant_closure)]
        pub fn scc(&self) -> Vec<Vec<usize>> {
            let ids = self.scc_ids();
            let group_num = ids.0;
            let mut counts = vec![0usize; group_num];
            for &x in ids.1.iter() {
                counts[x] += 1;
            }
            let mut groups: Vec<Vec<usize>> = (0..ids.0).map(|_| vec![]).collect();
            for i in 0..group_num {
                groups[i].reserve(counts[i]);
            }
            for i in 0..self.n {
                groups[ids.1[i]].push(i);
            }
            groups
        }
    }
}
pub mod internal_type_traits {

    use std::{
        fmt,
        iter::{Product, Sum},
        ops::{
            Add, AddAssign, BitAnd, BitAndAssign, BitOr, BitOrAssign, BitXor, BitXorAssign, Div,
            DivAssign, Mul, MulAssign, Not, Rem, RemAssign, Shl, ShlAssign, Shr, ShrAssign, Sub,
            SubAssign,
        },
    };

    pub trait Integral:
        'static
        + Send
        + Sync
        + Copy
        + Ord
        + Not<Output = Self>
        + Add<Output = Self>
        + Sub<Output = Self>
        + Mul<Output = Self>
        + Div<Output = Self>
        + Rem<Output = Self>
        + AddAssign
        + SubAssign
        + MulAssign
        + DivAssign
        + RemAssign
        + Sum
        + Product
        + BitOr<Output = Self>
        + BitAnd<Output = Self>
        + BitXor<Output = Self>
        + BitOrAssign
        + BitAndAssign
        + BitXorAssign
        + Shl<Output = Self>
        + Shr<Output = Self>
        + ShlAssign
        + ShrAssign
        + fmt::Display
        + fmt::Debug
        + fmt::Binary
        + fmt::Octal
        + Zero
        + One
        + BoundedBelow
        + BoundedAbove
    {
    }

    pub trait Zero {
        fn zero() -> Self;
    }

    pub trait One {
        fn one() -> Self;
    }

    pub trait BoundedBelow {
        fn min_value() -> Self;
    }

    pub trait BoundedAbove {
        fn max_value() -> Self;
    }

    macro_rules! impl_integral {
    ($($ty:ty),*) => {
        $(
            impl Zero for $ty {
                #[inline]
                fn zero() -> Self {
                    0
                }
            }

            impl One for $ty {
                #[inline]
                fn one() -> Self {
                    1
                }
            }

            impl BoundedBelow for $ty {
                #[inline]
                fn min_value() -> Self {
                    Self::MIN
                }
            }

            impl BoundedAbove for $ty {
                #[inline]
                fn max_value() -> Self {
                    Self::MAX
                }
            }

            impl Integral for $ty {}
        )*
    };
}

    impl_integral!(i8, i16, i32, i64, i128, isize, u8, u16, u32, u64, u128, usize);
}
pub mod lazysegtree {
    use crate::internal_bit::ceil_pow2;
    use crate::Monoid;

    pub trait MapMonoid {
        type M: Monoid;
        type F: Clone;

        fn identity_element() -> <Self::M as Monoid>::S {
            Self::M::identity()
        }

        fn binary_operation(
            a: &<Self::M as Monoid>::S,
            b: &<Self::M as Monoid>::S,
        ) -> <Self::M as Monoid>::S {
            Self::M::binary_operation(a, b)
        }

        fn identity_map() -> Self::F;

        fn mapping(f: &Self::F, x: &<Self::M as Monoid>::S) -> <Self::M as Monoid>::S;

        fn composition(f: &Self::F, g: &Self::F) -> Self::F;
    }

    impl<F: MapMonoid> Default for LazySegtree<F> {
        fn default() -> Self {
            Self::new(0)
        }
    }

    impl<F: MapMonoid> LazySegtree<F> {
        pub fn new(n: usize) -> Self {
            vec![F::identity_element(); n].into()
        }
    }

    impl<F: MapMonoid> From<Vec<<F::M as Monoid>::S>> for LazySegtree<F> {
        fn from(v: Vec<<F::M as Monoid>::S>) -> Self {
            let n = v.len();
            let log = ceil_pow2(n as u32) as usize;
            let size = 1 << log;
            let mut d = vec![F::identity_element(); 2 * size];
            let lz = vec![F::identity_map(); size];
            d[size..(size + n)].clone_from_slice(&v);
            let mut ret = LazySegtree {
                n,
                size,
                log,
                d,
                lz,
            };
            for i in (1..size).rev() {
                ret.update(i);
            }
            ret
        }
    }

    impl<F: MapMonoid> LazySegtree<F> {
        pub fn set(&mut self, mut p: usize, x: <F::M as Monoid>::S) {
            assert!(p < self.n);
            p += self.size;
            for i in (1..=self.log).rev() {
                self.push(p >> i);
            }
            self.d[p] = x;
            for i in 1..=self.log {
                self.update(p >> i);
            }
        }

        pub fn get(&mut self, mut p: usize) -> <F::M as Monoid>::S {
            assert!(p < self.n);
            p += self.size;
            for i in (1..=self.log).rev() {
                self.push(p >> i);
            }
            self.d[p].clone()
        }

        pub fn prod<R>(&mut self, range: R) -> <F::M as Monoid>::S
        where
            R: RangeBounds<usize>,
        {
            if range.start_bound() == Bound::Unbounded && range.end_bound() == Bound::Unbounded {
                return self.all_prod();
            }

            let mut r = match range.end_bound() {
                Bound::Included(r) => r + 1,
                Bound::Excluded(r) => *r,
                Bound::Unbounded => self.n,
            };
            let mut l = match range.start_bound() {
                Bound::Included(l) => *l,
                Bound::Excluded(l) => l + 1,
                Bound::Unbounded => 0,
            };

            assert!(l <= r && r <= self.n);
            if l == r {
                return F::identity_element();
            }

            l += self.size;
            r += self.size;

            for i in (1..=self.log).rev() {
                if ((l >> i) << i) != l {
                    self.push(l >> i);
                }
                if ((r >> i) << i) != r {
                    self.push(r >> i);
                }
            }

            let mut sml = F::identity_element();
            let mut smr = F::identity_element();
            while l < r {
                if l & 1 != 0 {
                    sml = F::binary_operation(&sml, &self.d[l]);
                    l += 1;
                }
                if r & 1 != 0 {
                    r -= 1;
                    smr = F::binary_operation(&self.d[r], &smr);
                }
                l >>= 1;
                r >>= 1;
            }

            F::binary_operation(&sml, &smr)
        }

        pub fn all_prod(&self) -> <F::M as Monoid>::S {
            self.d[1].clone()
        }

        pub fn apply(&mut self, mut p: usize, f: F::F) {
            assert!(p < self.n);
            p += self.size;
            for i in (1..=self.log).rev() {
                self.push(p >> i);
            }
            self.d[p] = F::mapping(&f, &self.d[p]);
            for i in 1..=self.log {
                self.update(p >> i);
            }
        }

        pub fn apply_range<R>(&mut self, range: R, f: F::F)
        where
            R: RangeBounds<usize>,
        {
            let mut r = match range.end_bound() {
                Bound::Included(r) => r + 1,
                Bound::Excluded(r) => *r,
                Bound::Unbounded => self.n,
            };
            let mut l = match range.start_bound() {
                Bound::Included(l) => *l,
                Bound::Excluded(l) => l + 1,
                Bound::Unbounded => 0,
            };

            assert!(l <= r && r <= self.n);
            if l == r {
                return;
            }

            l += self.size;
            r += self.size;

            for i in (1..=self.log).rev() {
                if ((l >> i) << i) != l {
                    self.push(l >> i);
                }
                if ((r >> i) << i) != r {
                    self.push((r - 1) >> i);
                }
            }

            {
                let l2 = l;
                let r2 = r;
                while l < r {
                    if l & 1 != 0 {
                        self.all_apply(l, f.clone());
                        l += 1;
                    }
                    if r & 1 != 0 {
                        r -= 1;
                        self.all_apply(r, f.clone());
                    }
                    l >>= 1;
                    r >>= 1;
                }
                l = l2;
                r = r2;
            }

            for i in 1..=self.log {
                if ((l >> i) << i) != l {
                    self.update(l >> i);
                }
                if ((r >> i) << i) != r {
                    self.update((r - 1) >> i);
                }
            }
        }

        pub fn max_right<G>(&mut self, mut l: usize, g: G) -> usize
        where
            G: Fn(<F::M as Monoid>::S) -> bool,
        {
            assert!(l <= self.n);
            assert!(g(F::identity_element()));
            if l == self.n {
                return self.n;
            }
            l += self.size;
            for i in (1..=self.log).rev() {
                self.push(l >> i);
            }
            let mut sm = F::identity_element();
            while {
                while l.is_multiple_of(2) {
                    l >>= 1;
                }
                if !g(F::binary_operation(&sm, &self.d[l])) {
                    while l < self.size {
                        self.push(l);
                        l *= 2;
                        let res = F::binary_operation(&sm, &self.d[l]);
                        if g(res.clone()) {
                            sm = res;
                            l += 1;
                        }
                    }
                    return l - self.size;
                }
                sm = F::binary_operation(&sm, &self.d[l]);
                l += 1;
                {
                    let l = l as isize;
                    (l & -l) != l
                }
            } {}
            self.n
        }

        pub fn min_left<G>(&mut self, mut r: usize, g: G) -> usize
        where
            G: Fn(<F::M as Monoid>::S) -> bool,
        {
            assert!(r <= self.n);
            assert!(g(F::identity_element()));
            if r == 0 {
                return 0;
            }
            r += self.size;
            for i in (1..=self.log).rev() {
                self.push((r - 1) >> i);
            }
            let mut sm = F::identity_element();
            while {
                r -= 1;
                while r > 1 && !r.is_multiple_of(2) {
                    r >>= 1;
                }
                if !g(F::binary_operation(&self.d[r], &sm)) {
                    while r < self.size {
                        self.push(r);
                        r = 2 * r + 1;
                        let res = F::binary_operation(&self.d[r], &sm);
                        if g(res.clone()) {
                            sm = res;
                            r -= 1;
                        }
                    }
                    return r + 1 - self.size;
                }
                sm = F::binary_operation(&self.d[r], &sm);
                {
                    let r = r as isize;
                    (r & -r) != r
                }
            } {}
            0
        }
    }

    #[derive(Clone)]
    pub struct LazySegtree<F>
    where
        F: MapMonoid,
    {
        n: usize,
        size: usize,
        log: usize,
        d: Vec<<F::M as Monoid>::S>,
        lz: Vec<F::F>,
    }

    impl<F> LazySegtree<F>
    where
        F: MapMonoid,
    {
        fn update(&mut self, k: usize) {
            self.d[k] = F::binary_operation(&self.d[2 * k], &self.d[2 * k + 1]);
        }

        fn all_apply(&mut self, k: usize, f: F::F) {
            self.d[k] = F::mapping(&f, &self.d[k]);
            if k < self.size {
                self.lz[k] = F::composition(&f, &self.lz[k]);
            }
        }

        fn push(&mut self, k: usize) {
            self.all_apply(2 * k, self.lz[k].clone());
            self.all_apply(2 * k + 1, self.lz[k].clone());
            self.lz[k] = F::identity_map();
        }
    }

    use std::{
        fmt::{Debug, Error, Formatter, Write},
        ops::{Bound, RangeBounds},
    };

    impl<F> Debug for LazySegtree<F>
    where
        F: MapMonoid,
        F::F: Debug,
        <F::M as Monoid>::S: Debug,
    {
        fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), Error> {
            for i in 0..self.log {
                for j in 0..1 << i {
                    f.write_fmt(format_args!(
                        "{:?}[{:?}]\t",
                        self.d[(1 << i) + j],
                        self.lz[(1 << i) + j]
                    ))?;
                }
                f.write_char('\n')?;
            }
            for i in 0..self.size {
                f.write_fmt(format_args!("{:?}\t", self.d[self.size + i]))?;
            }
            Ok(())
        }
    }
}
pub mod lca {
    #![warn(missing_docs)]

    use std::collections::VecDeque;
    use std::error::Error;
    use std::fmt;

    #[derive(Clone, Debug, Eq, PartialEq)]
    pub enum LcaError {
        InvalidVertex(usize),
        NotATree,
    }
    impl fmt::Display for LcaError {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "{self:?}")
        }
    }
    impl Error for LcaError {}

    #[derive(Clone, Debug)]
    pub struct Lca {
        parent: Vec<Vec<usize>>,
        depth: Vec<usize>,
    }

    impl Lca {
        pub fn new(
            vertex_count: usize,
            edges: &[(usize, usize)],
            root: usize,
        ) -> Result<Self, LcaError> {
            if root >= vertex_count {
                return Err(LcaError::InvalidVertex(root));
            }
            if edges.len() != vertex_count.saturating_sub(1) {
                return Err(LcaError::NotATree);
            }
            let mut graph = vec![Vec::new(); vertex_count];
            for &(a, b) in edges {
                if a >= vertex_count {
                    return Err(LcaError::InvalidVertex(a));
                }
                if b >= vertex_count {
                    return Err(LcaError::InvalidVertex(b));
                }
                graph[a].push(b);
                graph[b].push(a);
            }
            let levels = (usize::BITS - vertex_count.max(1).leading_zeros()) as usize;
            let mut parent = vec![vec![root; vertex_count]; levels];
            let mut depth = vec![0; vertex_count];
            let mut seen = vec![false; vertex_count];
            let mut queue = VecDeque::from([root]);
            seen[root] = true;
            while let Some(from) = queue.pop_front() {
                for &to in &graph[from] {
                    if seen[to] {
                        if parent[0][from] != to {
                            return Err(LcaError::NotATree);
                        }
                        continue;
                    }
                    seen[to] = true;
                    parent[0][to] = from;
                    depth[to] = depth[from] + 1;
                    queue.push_back(to);
                }
            }
            if seen.iter().any(|&x| !x) {
                return Err(LcaError::NotATree);
            }
            for level in 1..levels {
                for v in 0..vertex_count {
                    parent[level][v] = parent[level - 1][parent[level - 1][v]];
                }
            }
            Ok(Self { parent, depth })
        }
        pub fn lca(&self, mut a: usize, mut b: usize) -> Result<usize, LcaError> {
            self.check(a)?;
            self.check(b)?;
            if self.depth[a] > self.depth[b] {
                std::mem::swap(&mut a, &mut b);
            }
            b = self.kth_ancestor(b, self.depth[b] - self.depth[a]).unwrap();
            if a == b {
                return Ok(a);
            }
            for level in (0..self.parent.len()).rev() {
                if self.parent[level][a] != self.parent[level][b] {
                    a = self.parent[level][a];
                    b = self.parent[level][b];
                }
            }
            Ok(self.parent[0][a])
        }
        pub fn depth(&self, vertex: usize) -> Result<usize, LcaError> {
            self.check(vertex)?;
            Ok(self.depth[vertex])
        }
        pub fn distance(&self, a: usize, b: usize) -> Result<usize, LcaError> {
            let p = self.lca(a, b)?;
            Ok(self.depth[a] + self.depth[b] - 2 * self.depth[p])
        }
        pub fn kth_ancestor(&self, mut vertex: usize, steps: usize) -> Option<usize> {
            if vertex >= self.depth.len() || steps > self.depth[vertex] {
                return None;
            }
            for level in 0..self.parent.len() {
                if steps >> level & 1 != 0 {
                    vertex = self.parent[level][vertex];
                }
            }
            Some(vertex)
        }
        fn check(&self, v: usize) -> Result<(), LcaError> {
            if v < self.depth.len() {
                Ok(())
            } else {
                Err(LcaError::InvalidVertex(v))
            }
        }
    }
}
pub mod math {

    use crate::internal_math;

    use std::mem::swap;

    #[allow(clippy::many_single_char_names)]
    pub fn pow_mod(x: i64, mut n: i64, m: u32) -> u32 {
        assert!(0 <= n && 1 <= m && m <= 2u32.pow(31));
        if m == 1 {
            return 0;
        }
        let bt = internal_math::Barrett::new(m);
        let mut r = 1;
        let mut y = internal_math::safe_mod(x, m as i64) as u32;
        while n != 0 {
            if n & 1 != 0 {
                r = bt.mul(r, y);
            }
            y = bt.mul(y, y);
            n >>= 1;
        }
        r
    }

    pub fn inv_mod(x: i64, m: i64) -> i64 {
        assert!(1 <= m);
        let z = internal_math::inv_gcd(x, m);
        assert!(z.0 == 1);
        z.1
    }

    pub fn crt(r: &[i64], m: &[i64]) -> (i64, i64) {
        assert_eq!(r.len(), m.len());
        let (mut r0, mut m0) = (0, 1);
        for (&(mut ri), &(mut mi)) in r.iter().zip(m.iter()) {
            assert!(1 <= mi);
            ri = internal_math::safe_mod(ri, mi);
            if m0 < mi {
                swap(&mut r0, &mut ri);
                swap(&mut m0, &mut mi);
            }
            if m0 % mi == 0 {
                if r0 % mi != ri {
                    return (0, 0);
                }
                continue;
            }

            let (g, im) = internal_math::inv_gcd(m0, mi);
            let u1 = mi / g;
            if (ri - r0) % g != 0 {
                return (0, 0);
            }
            let x = (ri - r0) / g % u1 * im % u1;

            r0 += x * m0;
            m0 *= u1; // -> lcm(m0, mi)
            if r0 < 0 {
                r0 += m0
            };
        }

        (r0, m0)
    }

    #[allow(clippy::many_single_char_names)]
    pub fn floor_sum(n: i64, m: i64, a: i64, b: i64) -> i64 {
        use std::num::Wrapping as W;
        assert!((0..1i64 << 32).contains(&n));
        assert!((1..1i64 << 32).contains(&m));
        let mut ans = W(0_u64);
        let (wn, wm, mut wa, mut wb) = (W(n as u64), W(m as u64), W(a as u64), W(b as u64));
        if a < 0 {
            let a2 = W(internal_math::safe_mod(a, m) as u64);
            ans -= wn * (wn - W(1)) / W(2) * ((a2 - wa) / wm);
            wa = a2;
        }
        if b < 0 {
            let b2 = W(internal_math::safe_mod(b, m) as u64);
            ans -= wn * ((b2 - wb) / wm);
            wb = b2;
        }
        let ret = ans + internal_math::floor_sum_unsigned(wn, wm, wa, wb);
        ret.0 as i64
    }
}
pub mod matrix {
    #![warn(missing_docs)]

    use std::error::Error;
    use std::fmt;
    use std::ops::{Add, AddAssign, Index, IndexMut, Mul};

    #[derive(Clone, Debug, Eq, PartialEq)]
    pub enum MatrixError {
        RaggedRows,
        DimensionMismatch,
        NotSquare,
    }
    impl fmt::Display for MatrixError {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "{self:?}")
        }
    }
    impl Error for MatrixError {}

    #[derive(Clone, Debug, PartialEq, Eq)]
    pub struct Matrix<T> {
        rows: usize,
        cols: usize,
        data: Vec<T>,
    }
    impl<T: Clone> Matrix<T> {
        #[must_use]
        pub fn new(rows: usize, cols: usize, value: T) -> Self {
            Self {
                rows,
                cols,
                data: vec![value; rows.checked_mul(cols).expect("matrix size overflow")],
            }
        }
        pub fn from_rows(rows: Vec<Vec<T>>) -> Result<Self, MatrixError> {
            let row_count = rows.len();
            let cols = rows.first().map_or(0, Vec::len);
            if rows.iter().any(|r| r.len() != cols) {
                return Err(MatrixError::RaggedRows);
            }
            Ok(Self {
                rows: row_count,
                cols,
                data: rows.into_iter().flatten().collect(),
            })
        }
        pub fn transpose(&self) -> Self {
            if self.is_empty() {
                return Self {
                    rows: self.cols,
                    cols: self.rows,
                    data: Vec::new(),
                };
            }
            let mut data = Vec::with_capacity(self.data.len());
            for col in 0..self.cols {
                for row in 0..self.rows {
                    data.push(self[(row, col)].clone());
                }
            }
            Self {
                rows: self.cols,
                cols: self.rows,
                data,
            }
        }
    }
    impl<T> Matrix<T> {
        #[must_use]
        pub fn rows(&self) -> usize {
            self.rows
        }
        #[must_use]
        pub fn cols(&self) -> usize {
            self.cols
        }
        #[must_use]
        pub fn is_empty(&self) -> bool {
            self.rows == 0 || self.cols == 0
        }
        pub fn swap_rows(&mut self, a: usize, b: usize) {
            assert!(a < self.rows && b < self.rows);
            for c in 0..self.cols {
                self.data.swap(a * self.cols + c, b * self.cols + c);
            }
        }
        pub fn swap_columns(&mut self, a: usize, b: usize) {
            assert!(a < self.cols && b < self.cols);
            for r in 0..self.rows {
                self.data.swap(r * self.cols + a, r * self.cols + b);
            }
        }
    }
    impl<T> Index<(usize, usize)> for Matrix<T> {
        type Output = T;
        fn index(&self, (r, c): (usize, usize)) -> &T {
            assert!(r < self.rows && c < self.cols);
            &self.data[r * self.cols + c]
        }
    }
    impl<T> IndexMut<(usize, usize)> for Matrix<T> {
        fn index_mut(&mut self, (r, c): (usize, usize)) -> &mut T {
            assert!(r < self.rows && c < self.cols);
            &mut self.data[r * self.cols + c]
        }
    }
    impl<T> Index<usize> for Matrix<T> {
        type Output = [T];
        fn index(&self, r: usize) -> &[T] {
            assert!(r < self.rows);
            &self.data[r * self.cols..(r + 1) * self.cols]
        }
    }
    impl<T> IndexMut<usize> for Matrix<T> {
        fn index_mut(&mut self, r: usize) -> &mut [T] {
            assert!(r < self.rows);
            &mut self.data[r * self.cols..(r + 1) * self.cols]
        }
    }
    impl<T: fmt::Display> fmt::Display for Matrix<T> {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            for r in 0..self.rows {
                if r > 0 {
                    writeln!(f)?;
                }
                for c in 0..self.cols {
                    if c > 0 {
                        write!(f, " ")?;
                    }
                    write!(f, "{}", self[(r, c)])?;
                }
            }
            Ok(())
        }
    }
    impl<T> Matrix<T>
    where
        T: Copy + Default + AddAssign + Mul<Output = T>,
    {
        pub fn checked_mul(&self, rhs: &Self) -> Result<Self, MatrixError> {
            if self.cols != rhs.rows {
                return Err(MatrixError::DimensionMismatch);
            }
            let mut out = Self::new(self.rows, rhs.cols, T::default());
            if out.is_empty() || self.cols == 0 {
                return Ok(out);
            }
            for i in 0..self.rows {
                for k in 0..self.cols {
                    for j in 0..rhs.cols {
                        out[(i, j)] += self[(i, k)] * rhs[(k, j)];
                    }
                }
            }
            Ok(out)
        }
        pub fn pow(&self, mut exponent: u64, one: T) -> Result<Self, MatrixError> {
            if self.rows != self.cols {
                return Err(MatrixError::NotSquare);
            }
            let mut result = Self::new(self.rows, self.cols, T::default());
            for i in 0..self.rows {
                result[(i, i)] = one;
            }
            let mut base = self.clone();
            while exponent > 0 {
                if exponent & 1 != 0 {
                    result = result.checked_mul(&base)?;
                }
                exponent >>= 1;
                if exponent > 0 {
                    base = base.checked_mul(&base)?;
                }
            }
            Ok(result)
        }
    }
    impl<T> Mul for &Matrix<T>
    where
        T: Copy + Default + AddAssign + Mul<Output = T>,
    {
        type Output = Matrix<T>;
        fn mul(self, rhs: Self) -> Self::Output {
            self.checked_mul(rhs).expect("matrix dimension mismatch")
        }
    }
    impl<T> Add for &Matrix<T>
    where
        T: Copy + Add<Output = T>,
    {
        type Output = Matrix<T>;
        fn add(self, rhs: Self) -> Self::Output {
            assert_eq!((self.rows, self.cols), (rhs.rows, rhs.cols));
            Matrix {
                rows: self.rows,
                cols: self.cols,
                data: self
                    .data
                    .iter()
                    .zip(&rhs.data)
                    .map(|(&a, &b)| a + b)
                    .collect(),
            }
        }
    }
}
pub mod maxflow {

    #![allow(dead_code, clippy::bool_to_int_with_if)]
    use crate::internal_queue::SimpleQueue;
    use crate::internal_type_traits::Integral;
    use std::cmp::min;
    use std::iter;

    impl<Cap> MfGraph<Cap>
    where
        Cap: Integral,
    {
        pub fn new(n: usize) -> MfGraph<Cap> {
            MfGraph {
                _n: n,
                pos: Vec::new(),
                g: iter::repeat_with(Vec::new).take(n).collect(),
            }
        }

        pub fn add_edge(&mut self, from: usize, to: usize, cap: Cap) -> usize {
            assert!(from < self._n);
            assert!(to < self._n);
            assert!(Cap::zero() <= cap);
            let m = self.pos.len();
            self.pos.push((from, self.g[from].len()));
            let rev = self.g[to].len() + usize::from(from == to);
            self.g[from].push(_Edge { to, rev, cap });
            let rev = self.g[from].len() - 1;
            self.g[to].push(_Edge {
                to: from,
                rev,
                cap: Cap::zero(),
            });
            m
        }
    }

    #[derive(Clone, Debug, PartialEq, Eq)]
    pub struct Edge<Cap: Integral> {
        pub from: usize,
        pub to: usize,
        pub cap: Cap,
        pub flow: Cap,
    }

    impl<Cap> MfGraph<Cap>
    where
        Cap: Integral,
    {
        pub fn get_edge(&self, i: usize) -> Edge<Cap> {
            let m = self.pos.len();
            assert!(i < m);
            let _e = &self.g[self.pos[i].0][self.pos[i].1];
            let _re = &self.g[_e.to][_e.rev];
            Edge {
                from: self.pos[i].0,
                to: _e.to,
                cap: _e.cap + _re.cap,
                flow: _re.cap,
            }
        }
        pub fn edges(&self) -> Vec<Edge<Cap>> {
            let m = self.pos.len();
            (0..m).map(|i| self.get_edge(i)).collect()
        }
        pub fn change_edge(&mut self, i: usize, new_cap: Cap, new_flow: Cap) {
            let m = self.pos.len();
            assert!(i < m);
            assert!(Cap::zero() <= new_flow && new_flow <= new_cap);
            let (to, rev) = {
                let _e = &mut self.g[self.pos[i].0][self.pos[i].1];
                _e.cap = new_cap - new_flow;
                (_e.to, _e.rev)
            };
            let _re = &mut self.g[to][rev];
            _re.cap = new_flow;
        }

        pub fn flow(&mut self, s: usize, t: usize) -> Cap {
            self.flow_with_capacity(s, t, Cap::max_value())
        }
        pub fn flow_with_capacity(&mut self, s: usize, t: usize, flow_limit: Cap) -> Cap {
            let n_ = self._n;
            assert!(s < n_);
            assert!(t < n_);
            assert_ne!(s, t);
            assert!(Cap::zero() <= flow_limit);

            let mut calc = FlowCalculator {
                graph: self,
                s,
                t,
                flow_limit,
                level: vec![0; n_],
                iter: vec![0; n_],
                que: SimpleQueue::default(),
            };

            let mut flow = Cap::zero();
            while flow < flow_limit {
                calc.bfs();
                if calc.level[t] == -1 {
                    break;
                }
                calc.iter.iter_mut().for_each(|e| *e = 0);
                while flow < flow_limit {
                    let f = calc.dfs(t, flow_limit - flow);
                    if f == Cap::zero() {
                        break;
                    }
                    flow += f;
                }
            }
            flow
        }

        pub fn min_cut(&self, s: usize) -> Vec<bool> {
            let mut visited = vec![false; self._n];
            let mut que = SimpleQueue::default();
            que.push(s);
            while let Some(&p) = que.pop() {
                visited[p] = true;
                for e in &self.g[p] {
                    if e.cap != Cap::zero() && !visited[e.to] {
                        visited[e.to] = true;
                        que.push(e.to);
                    }
                }
            }
            visited
        }
    }

    struct FlowCalculator<'a, Cap> {
        graph: &'a mut MfGraph<Cap>,
        s: usize,
        t: usize,
        flow_limit: Cap,
        level: Vec<i32>,
        iter: Vec<usize>,
        que: SimpleQueue<usize>,
    }

    impl<Cap> FlowCalculator<'_, Cap>
    where
        Cap: Integral,
    {
        fn bfs(&mut self) {
            self.level.iter_mut().for_each(|e| *e = -1);
            self.level[self.s] = 0;
            self.que.clear();
            self.que.push(self.s);
            while let Some(&v) = self.que.pop() {
                for e in &self.graph.g[v] {
                    if e.cap == Cap::zero() || self.level[e.to] >= 0 {
                        continue;
                    }
                    self.level[e.to] = self.level[v] + 1;
                    if e.to == self.t {
                        return;
                    }
                    self.que.push(e.to);
                }
            }
        }
        fn dfs(&mut self, v: usize, up: Cap) -> Cap {
            if v == self.s {
                return up;
            }
            let mut res = Cap::zero();
            let level_v = self.level[v];
            for i in self.iter[v]..self.graph.g[v].len() {
                self.iter[v] = i;
                let &_Edge {
                    to: e_to,
                    rev: e_rev,
                    ..
                } = &self.graph.g[v][i];
                if level_v <= self.level[e_to] || self.graph.g[e_to][e_rev].cap == Cap::zero() {
                    continue;
                }
                let d = self.dfs(e_to, min(up - res, self.graph.g[e_to][e_rev].cap));
                if d <= Cap::zero() {
                    continue;
                }
                self.graph.g[v][i].cap += d;
                self.graph.g[e_to][e_rev].cap -= d;
                res += d;
                if res == up {
                    return res;
                }
            }
            self.iter[v] = self.graph.g[v].len();
            res
        }
    }

    #[derive(Clone, Debug, Default)]
    pub struct MfGraph<Cap> {
        _n: usize,
        pos: Vec<(usize, usize)>,
        g: Vec<Vec<_Edge<Cap>>>,
    }

    #[derive(Clone, Debug)]
    struct _Edge<Cap> {
        to: usize,
        rev: usize,
        cap: Cap,
    }
}
pub mod mex {
    #![warn(missing_docs)]

    use std::collections::BTreeMap;
    use std::ops::{Bound, RangeBounds};

    const DOMAIN_START: i128 = i64::MIN as i128;
    const DOMAIN_END: i128 = i64::MAX as i128 + 1;

    fn bounds<R: RangeBounds<i64>>(range: R) -> (i128, i128) {
        let start = match range.start_bound() {
            Bound::Included(&x) => x as i128,
            Bound::Excluded(&x) => x as i128 + 1,
            Bound::Unbounded => DOMAIN_START,
        };
        let end = match range.end_bound() {
            Bound::Included(&x) => x as i128 + 1,
            Bound::Excluded(&x) => x as i128,
            Bound::Unbounded => DOMAIN_END,
        };
        assert!(start <= end, "range is reversed");
        (start, end)
    }

    #[derive(Clone, Debug, Default)]
    pub struct Mex {
        intervals: BTreeMap<i128, i128>,
    }
    impl Mex {
        #[must_use]
        pub fn new() -> Self {
            Self::default()
        }
        #[must_use]
        pub fn contains(&self, value: i64) -> bool {
            let x = value as i128;
            self.intervals
                .range(..=x)
                .next_back()
                .is_some_and(|(_, &end)| x < end)
        }
        pub fn insert(&mut self, value: i64) -> bool {
            self.insert_range(value..=value)
        }
        pub fn remove(&mut self, value: i64) -> bool {
            self.remove_range(value..=value)
        }
        pub fn insert_range<R: RangeBounds<i64>>(&mut self, range: R) -> bool {
            let (mut start, mut end) = bounds(range);
            if start == end {
                return false;
            }
            if self.contains_all_i128(start, end) {
                return false;
            }

            if let Some((&key, &old_end)) = self.intervals.range(..=start).next_back() {
                if old_end >= start {
                    self.intervals.remove(&key);
                    start = key;
                    end = end.max(old_end);
                }
            }
            let keys = self
                .intervals
                .range(start..=end)
                .map(|(&interval_start, _)| interval_start)
                .collect::<Vec<_>>();
            for key in keys {
                let old_end = self.intervals.remove(&key).unwrap();
                end = end.max(old_end);
            }
            self.intervals.insert(start, end);
            true
        }
        pub fn remove_range<R: RangeBounds<i64>>(&mut self, range: R) -> bool {
            let (start, end) = bounds(range);
            if start == end {
                return false;
            }
            let mut keys = Vec::new();
            if let Some((&key, &old_end)) = self.intervals.range(..start).next_back() {
                if old_end > start {
                    keys.push(key);
                }
            }
            keys.extend(
                self.intervals
                    .range(start..end)
                    .map(|(&interval_start, _)| interval_start),
            );
            if keys.is_empty() {
                return false;
            }
            for key in keys {
                let old_end = self.intervals.remove(&key).unwrap();
                if key < start {
                    self.intervals.insert(key, start);
                }
                if old_end > end {
                    self.intervals.insert(end, old_end);
                }
            }
            true
        }
        #[must_use]
        pub fn contains_any<R: RangeBounds<i64>>(&self, range: R) -> bool {
            let (start, end) = bounds(range);
            start < end
                && self
                    .intervals
                    .range(..end)
                    .next_back()
                    .is_some_and(|(_, &e)| e > start)
        }
        #[must_use]
        pub fn contains_all<R: RangeBounds<i64>>(&self, range: R) -> bool {
            let (start, end) = bounds(range);
            self.contains_all_i128(start, end)
        }
        fn contains_all_i128(&self, start: i128, end: i128) -> bool {
            start == end
                || self
                    .intervals
                    .range(..=start)
                    .next_back()
                    .is_some_and(|(_, &e)| e >= end)
        }
        #[must_use]
        pub fn mex<R: RangeBounds<i64>>(&self, range: R) -> Option<i64> {
            let (start, end) = bounds(range);
            if start == end {
                return None;
            }
            let candidate = self
                .intervals
                .range(..=start)
                .next_back()
                .map_or(
                    start,
                    |(&s, &e)| if s <= start && start < e { e } else { start },
                );
            (candidate < end).then_some(candidate as i64)
        }
    }
}
pub mod mincostflow {

    use crate::internal_type_traits::Integral;

    #[derive(Clone, Debug)]
    pub struct MinCostFlowEdge<T> {
        pub from: usize,
        pub to: usize,
        pub cap: T,
        pub flow: T,
        pub cost: T,
    }

    #[derive(Clone, Debug)]
    pub struct MinCostFlowGraph<T> {
        pos: Vec<(usize, usize)>,
        g: Vec<Vec<_Edge<T>>>,
    }

    #[allow(clippy::redundant_closure)]
    impl<T> MinCostFlowGraph<T>
    where
        T: Integral + std::ops::Neg<Output = T>,
    {
        pub fn new(n: usize) -> Self {
            Self {
                pos: vec![],
                g: (0..n).map(|_| vec![]).collect(),
            }
        }

        pub fn get_edge(&self, i: usize) -> MinCostFlowEdge<T> {
            assert!(i < self.pos.len());
            let e = &self.g[self.pos[i].0][self.pos[i].1];
            let re = &self.g[e.to][e.rev];
            MinCostFlowEdge {
                from: self.pos[i].0,
                to: e.to,
                cap: e.cap + re.cap,
                flow: re.cap,
                cost: e.cost,
            }
        }

        pub fn edges(&self) -> Vec<MinCostFlowEdge<T>> {
            let m = self.pos.len();
            let mut result = vec![];
            for i in 0..m {
                result.push(self.get_edge(i));
            }
            result
        }

        pub fn add_edge(&mut self, from: usize, to: usize, cap: T, cost: T) -> usize {
            assert!(from < self.g.len());
            assert!(to < self.g.len());
            assert_ne!(from, to);
            assert!(cap >= T::zero());
            assert!(cost >= T::zero());

            self.pos.push((from, self.g[from].len()));

            let rev = self.g[to].len();
            self.g[from].push(_Edge { to, rev, cap, cost });

            let rev = self.g[from].len() - 1;
            self.g[to].push(_Edge {
                to: from,
                rev,
                cap: T::zero(),
                cost: -cost,
            });

            self.pos.len() - 1
        }

        pub fn flow(&mut self, source: usize, sink: usize, flow_limit: T) -> (T, T) {
            self.slope(source, sink, flow_limit).pop().unwrap()
        }

        pub fn slope(&mut self, source: usize, sink: usize, flow_limit: T) -> Vec<(T, T)> {
            let n = self.g.len();
            assert!(source < n);
            assert!(sink < n);
            assert_ne!(source, sink);

            let mut dual = vec![T::zero(); n];
            let mut prev_v = vec![0; n];
            let mut prev_e = vec![0; n];
            let mut flow = T::zero();
            let mut cost = T::zero();
            let mut prev_cost_per_flow: Option<T> = None;
            let mut result = vec![(flow, cost)];
            while flow < flow_limit {
                if !self.refine_dual(source, sink, &mut dual, &mut prev_v, &mut prev_e) {
                    break;
                }

                let mut c = flow_limit - flow;
                let mut v = sink;
                while v != source {
                    c = std::cmp::min(c, self.g[prev_v[v]][prev_e[v]].cap);
                    v = prev_v[v];
                }

                let mut v = sink;
                while v != source {
                    self.g[prev_v[v]][prev_e[v]].cap -= c;
                    let rev = self.g[prev_v[v]][prev_e[v]].rev;
                    self.g[v][rev].cap += c;
                    v = prev_v[v];
                }

                let d = -dual[source];
                flow += c;
                cost += d * c;
                if prev_cost_per_flow == Some(d) {
                    assert!(result.pop().is_some());
                }
                result.push((flow, cost));
                prev_cost_per_flow = Some(d);
            }
            result
        }

        fn refine_dual(
            &self,
            source: usize,
            sink: usize,
            dual: &mut [T],
            pv: &mut [usize],
            pe: &mut [usize],
        ) -> bool {
            let n = self.g.len();
            let mut dist = vec![T::max_value(); n];
            let mut vis = vec![false; n];

            let mut que = std::collections::BinaryHeap::new();
            dist[source] = T::zero();
            que.push((std::cmp::Reverse(T::zero()), source));
            while let Some((_, v)) = que.pop() {
                if vis[v] {
                    continue;
                }
                vis[v] = true;
                if v == sink {
                    break;
                }

                for (i, e) in self.g[v].iter().enumerate() {
                    if vis[e.to] || e.cap == T::zero() {
                        continue;
                    }

                    let cost = e.cost - dual[e.to] + dual[v];
                    if dist[e.to] - dist[v] > cost {
                        dist[e.to] = dist[v] + cost;
                        pv[e.to] = v;
                        pe[e.to] = i;
                        que.push((std::cmp::Reverse(dist[e.to]), e.to));
                    }
                }
            }

            if !vis[sink] {
                return false;
            }

            for v in 0..n {
                if !vis[v] {
                    continue;
                }
                dual[v] -= dist[sink] - dist[v];
            }
            true
        }
    }

    #[derive(Clone, Debug)]
    struct _Edge<T> {
        to: usize,
        rev: usize,
        cap: T,
        cost: T,
    }
}
pub mod modint {

    use crate::internal_math;
    use std::{
        cell::RefCell,
        convert::{Infallible, TryInto as _},
        fmt,
        hash::{Hash, Hasher},
        iter::{Product, Sum},
        marker::PhantomData,
        ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign},
        str::FromStr,
        sync::atomic::{self, AtomicU32, AtomicU64},
        thread::LocalKey,
    };

    pub type ModInt1000000007 = StaticModInt<Mod1000000007>;
    pub type ModInt998244353 = StaticModInt<Mod998244353>;
    pub type ModInt = DynamicModInt<DefaultId>;

    #[derive(Copy, Clone, Eq, PartialEq)]
    #[repr(transparent)]
    pub struct StaticModInt<M> {
        val: u32,
        phantom: PhantomData<fn() -> M>,
    }

    impl<M: Modulus> StaticModInt<M> {
        #[inline(always)]
        pub fn modulus() -> u32 {
            M::VALUE
        }

        #[inline]
        pub fn new<T: RemEuclidU32>(val: T) -> Self {
            Self::raw(val.rem_euclid_u32(M::VALUE))
        }

        #[inline]
        pub fn raw(val: u32) -> Self {
            Self {
                val,
                phantom: PhantomData,
            }
        }

        #[inline]
        pub fn val(self) -> u32 {
            self.val
        }

        #[inline]
        pub fn pow(self, n: u64) -> Self {
            <Self as ModIntBase>::pow(self, n)
        }

        #[inline]
        pub fn inv(self) -> Self {
            if M::HINT_VALUE_IS_PRIME {
                if self.val() == 0 {
                    panic!("attempt to divide by zero");
                }
                debug_assert!(
                    internal_math::is_prime(M::VALUE.try_into().unwrap()),
                    "{} is not a prime number",
                    M::VALUE,
                );
                self.pow((M::VALUE - 2).into())
            } else {
                Self::inv_for_non_prime_modulus(self)
            }
        }
    }

    impl<M: Modulus> ModIntBase for StaticModInt<M> {
        #[inline(always)]
        fn modulus() -> u32 {
            Self::modulus()
        }

        #[inline]
        fn raw(val: u32) -> Self {
            Self::raw(val)
        }

        #[inline]
        fn val(self) -> u32 {
            self.val()
        }

        #[inline]
        fn inv(self) -> Self {
            self.inv()
        }
    }

    pub trait Modulus: 'static + Copy + Eq {
        const VALUE: u32;
        const HINT_VALUE_IS_PRIME: bool;

        fn butterfly_cache() -> &'static LocalKey<RefCell<Option<ButterflyCache<Self>>>>;
    }

    #[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
    pub enum Mod1000000007 {}

    impl Modulus for Mod1000000007 {
        const VALUE: u32 = 1_000_000_007;
        const HINT_VALUE_IS_PRIME: bool = true;

        fn butterfly_cache() -> &'static LocalKey<RefCell<Option<ButterflyCache<Self>>>> {
            thread_local! {
                static BUTTERFLY_CACHE: RefCell<Option<ButterflyCache<Mod1000000007>>> = RefCell::default();
            }
            &BUTTERFLY_CACHE
        }
    }

    #[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
    pub enum Mod998244353 {}

    impl Modulus for Mod998244353 {
        const VALUE: u32 = 998_244_353;
        const HINT_VALUE_IS_PRIME: bool = true;

        fn butterfly_cache() -> &'static LocalKey<RefCell<Option<ButterflyCache<Self>>>> {
            thread_local! {
                static BUTTERFLY_CACHE: RefCell<Option<ButterflyCache<Mod998244353>>> = RefCell::default();
            }
            &BUTTERFLY_CACHE
        }
    }

    pub struct ButterflyCache<M> {
        pub(crate) sum_e: Vec<StaticModInt<M>>,
        pub(crate) sum_ie: Vec<StaticModInt<M>>,
    }

    #[derive(Copy, Clone, Eq, PartialEq)]
    #[repr(transparent)]
    pub struct DynamicModInt<I> {
        val: u32,
        phantom: PhantomData<fn() -> I>,
    }

    impl<I: Id> DynamicModInt<I> {
        #[inline]
        pub fn modulus() -> u32 {
            I::companion_barrett().umod()
        }

        #[inline]
        pub fn set_modulus(modulus: u32) {
            if modulus == 0 {
                panic!("the modulus must not be 0");
            }
            I::companion_barrett().update(modulus);
        }

        #[inline]
        pub fn new<T: RemEuclidU32>(val: T) -> Self {
            <Self as ModIntBase>::new(val)
        }

        #[inline]
        pub fn raw(val: u32) -> Self {
            Self {
                val,
                phantom: PhantomData,
            }
        }

        #[inline]
        pub fn val(self) -> u32 {
            self.val
        }

        #[inline]
        pub fn pow(self, n: u64) -> Self {
            <Self as ModIntBase>::pow(self, n)
        }

        #[inline]
        pub fn inv(self) -> Self {
            Self::inv_for_non_prime_modulus(self)
        }
    }

    impl<I: Id> ModIntBase for DynamicModInt<I> {
        #[inline]
        fn modulus() -> u32 {
            Self::modulus()
        }

        #[inline]
        fn raw(val: u32) -> Self {
            Self::raw(val)
        }

        #[inline]
        fn val(self) -> u32 {
            self.val()
        }

        #[inline]
        fn inv(self) -> Self {
            self.inv()
        }
    }

    pub trait Id: 'static + Copy + Eq {
        fn companion_barrett() -> &'static Barrett;
    }

    #[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
    pub enum DefaultId {}

    impl Id for DefaultId {
        fn companion_barrett() -> &'static Barrett {
            static BARRETT: Barrett = Barrett::default();
            &BARRETT
        }
    }

    pub struct Barrett {
        m: AtomicU32,
        im: AtomicU64,
    }

    impl Barrett {
        #[inline]
        pub const fn new(m: u32) -> Self {
            Self {
                m: AtomicU32::new(m),
                im: AtomicU64::new((-1i64 as u64 / m as u64).wrapping_add(1)),
            }
        }

        #[inline]
        const fn default() -> Self {
            Self::new(998_244_353)
        }

        #[inline]
        fn update(&self, m: u32) {
            let im = (-1i64 as u64 / m as u64).wrapping_add(1);
            self.m.store(m, atomic::Ordering::SeqCst);
            self.im.store(im, atomic::Ordering::SeqCst);
        }

        #[inline]
        fn umod(&self) -> u32 {
            self.m.load(atomic::Ordering::SeqCst)
        }

        #[inline]
        fn mul(&self, a: u32, b: u32) -> u32 {
            let m = self.m.load(atomic::Ordering::SeqCst);
            let im = self.im.load(atomic::Ordering::SeqCst);
            internal_math::mul_mod(a, b, m, im)
        }
    }

    impl Default for Barrett {
        #[inline]
        fn default() -> Self {
            Self::default()
        }
    }

    pub trait ModIntBase:
        Default
        + FromStr
        + From<i8>
        + From<i16>
        + From<i32>
        + From<i64>
        + From<i128>
        + From<isize>
        + From<u8>
        + From<u16>
        + From<u32>
        + From<u64>
        + From<u128>
        + From<usize>
        + Copy
        + Eq
        + Hash
        + fmt::Display
        + fmt::Debug
        + Neg<Output = Self>
        + Add<Output = Self>
        + Sub<Output = Self>
        + Mul<Output = Self>
        + Div<Output = Self>
        + AddAssign
        + SubAssign
        + MulAssign
        + DivAssign
    {
        fn modulus() -> u32;

        fn raw(val: u32) -> Self;

        fn val(self) -> u32;

        fn inv(self) -> Self;

        #[inline]
        fn new<T: RemEuclidU32>(val: T) -> Self {
            Self::raw(val.rem_euclid_u32(Self::modulus()))
        }

        #[inline]
        fn pow(self, mut n: u64) -> Self {
            let mut x = self;
            let mut r = Self::raw(u32::from(Self::modulus() > 1));
            while n > 0 {
                if n & 1 == 1 {
                    r *= x;
                }
                x *= x;
                n >>= 1;
            }
            r
        }
    }

    pub trait RemEuclidU32 {
        fn rem_euclid_u32(self, modulus: u32) -> u32;
    }

    macro_rules! impl_rem_euclid_u32_for_small_signed {
    ($($ty:tt),*) => {
        $(
            impl RemEuclidU32 for $ty {
                #[inline]
                fn rem_euclid_u32(self, modulus: u32) -> u32 {
                    (self as i64).rem_euclid(i64::from(modulus)) as _
                }
            }
        )*
    }
}

    impl_rem_euclid_u32_for_small_signed!(i8, i16, i32, i64, isize);

    impl RemEuclidU32 for i128 {
        #[inline]
        fn rem_euclid_u32(self, modulus: u32) -> u32 {
            self.rem_euclid(i128::from(modulus)) as _
        }
    }

    macro_rules! impl_rem_euclid_u32_for_small_unsigned {
    ($($ty:tt),*) => {
        $(
            impl RemEuclidU32 for $ty {
                #[inline]
                fn rem_euclid_u32(self, modulus: u32) -> u32 {
                    self as u32 % modulus
                }
            }
        )*
    }
}

    macro_rules! impl_rem_euclid_u32_for_large_unsigned {
    ($($ty:tt),*) => {
        $(
            impl RemEuclidU32 for $ty {
                #[inline]
                fn rem_euclid_u32(self, modulus: u32) -> u32 {
                    (self % (modulus as $ty)) as _
                }
            }
        )*
    }
}

    impl_rem_euclid_u32_for_small_unsigned!(u8, u16, u32);
    impl_rem_euclid_u32_for_large_unsigned!(u64, u128);

    #[cfg(target_pointer_width = "32")]
    impl_rem_euclid_u32_for_small_unsigned!(usize);

    #[cfg(target_pointer_width = "64")]
    impl_rem_euclid_u32_for_large_unsigned!(usize);

    trait InternalImplementations: ModIntBase {
        #[inline]
        fn inv_for_non_prime_modulus(this: Self) -> Self {
            let (gcd, x) = internal_math::inv_gcd(this.val().into(), Self::modulus().into());
            if gcd != 1 {
                panic!("the multiplicative inverse does not exist");
            }
            Self::new(x)
        }

        #[inline]
        fn default_impl() -> Self {
            Self::raw(0)
        }

        #[inline]
        fn from_str_impl(s: &str) -> Result<Self, Infallible> {
            Ok(s.parse::<i64>()
                .map(Self::new)
                .unwrap_or_else(|_| todo!("parsing as an arbitrary precision integer?")))
        }

        #[inline]
        fn hash_impl(this: &Self, state: &mut impl Hasher) {
            this.val().hash(state)
        }

        #[inline]
        fn display_impl(this: &Self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            fmt::Display::fmt(&this.val(), f)
        }

        #[inline]
        fn debug_impl(this: &Self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            fmt::Debug::fmt(&this.val(), f)
        }

        #[inline]
        fn neg_impl(this: Self) -> Self {
            Self::sub_impl(Self::raw(0), this)
        }

        #[inline]
        fn add_impl(lhs: Self, rhs: Self) -> Self {
            let modulus = Self::modulus();
            let mut val = lhs.val() + rhs.val();
            if val >= modulus {
                val -= modulus;
            }
            Self::raw(val)
        }

        #[inline]
        fn sub_impl(lhs: Self, rhs: Self) -> Self {
            let modulus = Self::modulus();
            let mut val = lhs.val().wrapping_sub(rhs.val());
            if val >= modulus {
                val = val.wrapping_add(modulus)
            }
            Self::raw(val)
        }

        fn mul_impl(lhs: Self, rhs: Self) -> Self;

        #[inline]
        fn div_impl(lhs: Self, rhs: Self) -> Self {
            Self::mul_impl(lhs, rhs.inv())
        }
    }

    impl<M: Modulus> InternalImplementations for StaticModInt<M> {
        #[inline]
        fn mul_impl(lhs: Self, rhs: Self) -> Self {
            Self::raw((u64::from(lhs.val()) * u64::from(rhs.val()) % u64::from(M::VALUE)) as u32)
        }
    }

    impl<I: Id> InternalImplementations for DynamicModInt<I> {
        #[inline]
        fn mul_impl(lhs: Self, rhs: Self) -> Self {
            Self::raw(I::companion_barrett().mul(lhs.val, rhs.val))
        }
    }

    macro_rules! impl_basic_traits {
    () => {};
    (impl <$generic_param:ident : $generic_param_bound:tt> _ for $self:ty; $($rest:tt)*) => {
        impl <$generic_param: $generic_param_bound> Default for $self {
            #[inline]
            fn default() -> Self {
                Self::default_impl()
            }
        }

        impl <$generic_param: $generic_param_bound> FromStr for $self {
            type Err = Infallible;

            #[inline]
            fn from_str(s: &str) -> Result<Self, Infallible> {
                Self::from_str_impl(s)
            }
        }

        impl<$generic_param: $generic_param_bound, V: RemEuclidU32> From<V> for $self {
            #[inline]
            fn from(from: V) -> Self {
                Self::new(from)
            }
        }

        #[allow(clippy::derived_hash_with_manual_eq)]
        impl<$generic_param: $generic_param_bound> Hash for $self {
            #[inline]
            fn hash<H: Hasher>(&self, state: &mut H) {
                Self::hash_impl(self, state)
            }
        }

        impl<$generic_param: $generic_param_bound> fmt::Display for $self {
            #[inline]
            fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
                Self::display_impl(self, f)
            }
        }

        impl<$generic_param: $generic_param_bound> fmt::Debug for $self {
            #[inline]
            fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
                Self::debug_impl(self, f)
            }
        }

        impl<$generic_param: $generic_param_bound> Neg for $self {
            type Output = $self;

            #[inline]
            fn neg(self) -> $self {
                Self::neg_impl(self)
            }
        }

        impl<$generic_param: $generic_param_bound> Neg for &'_ $self {
            type Output = $self;

            #[inline]
            fn neg(self) -> $self {
                <$self>::neg_impl(*self)
            }
        }

        impl_basic_traits!($($rest)*);
    };
}

    impl_basic_traits! {
        impl <M: Modulus> _ for StaticModInt<M> ;
        impl <I: Id     > _ for DynamicModInt<I>;
    }

    macro_rules! impl_bin_ops {
    () => {};
    (for<$($generic_param:ident : $generic_param_bound:tt),*> <$lhs_ty:ty> ~ <$rhs_ty:ty> -> $output:ty { { $lhs_body:expr } ~ { $rhs_body:expr } } $($rest:tt)*) => {
        impl <$($generic_param: $generic_param_bound),*> Add<$rhs_ty> for $lhs_ty {
            type Output = $output;

            #[inline]
            fn add(self, rhs: $rhs_ty) -> $output {
                <$output>::add_impl(apply($lhs_body, self), apply($rhs_body, rhs))
            }
        }

        impl <$($generic_param: $generic_param_bound),*> Sub<$rhs_ty> for $lhs_ty {
            type Output = $output;

            #[inline]
            fn sub(self, rhs: $rhs_ty) -> $output {
                <$output>::sub_impl(apply($lhs_body, self), apply($rhs_body, rhs))
            }
        }

        impl <$($generic_param: $generic_param_bound),*> Mul<$rhs_ty> for $lhs_ty {
            type Output = $output;

            #[inline]
            fn mul(self, rhs: $rhs_ty) -> $output {
                <$output>::mul_impl(apply($lhs_body, self), apply($rhs_body, rhs))
            }
        }

        impl <$($generic_param: $generic_param_bound),*> Div<$rhs_ty> for $lhs_ty {
            type Output = $output;

            #[inline]
            fn div(self, rhs: $rhs_ty) -> $output {
                <$output>::div_impl(apply($lhs_body, self), apply($rhs_body, rhs))
            }
        }

        impl_bin_ops!($($rest)*);
    };
}

    macro_rules! impl_assign_ops {
    () => {};
    (for<$($generic_param:ident : $generic_param_bound:tt),*> <$lhs_ty:ty> ~= <$rhs_ty:ty> { _ ~= { $rhs_body:expr } } $($rest:tt)*) => {
        impl <$($generic_param: $generic_param_bound),*> AddAssign<$rhs_ty> for $lhs_ty {
            #[inline]
            fn add_assign(&mut self, rhs: $rhs_ty) {
                *self = *self + apply($rhs_body, rhs);
            }
        }

        impl <$($generic_param: $generic_param_bound),*> SubAssign<$rhs_ty> for $lhs_ty {
            #[inline]
            fn sub_assign(&mut self, rhs: $rhs_ty) {
                *self = *self - apply($rhs_body, rhs);
            }
        }

        impl <$($generic_param: $generic_param_bound),*> MulAssign<$rhs_ty> for $lhs_ty {
            #[inline]
            fn mul_assign(&mut self, rhs: $rhs_ty) {
                *self = *self * apply($rhs_body, rhs);
            }
        }

        impl <$($generic_param: $generic_param_bound),*> DivAssign<$rhs_ty> for $lhs_ty {
            #[inline]
            fn div_assign(&mut self, rhs: $rhs_ty) {
                *self = *self / apply($rhs_body, rhs);
            }
        }

        impl_assign_ops!($($rest)*);
    };
}

    #[inline]
    fn apply<F: FnOnce(X) -> O, X, O>(f: F, x: X) -> O {
        f(x)
    }

    impl_bin_ops! {
        for<M: Modulus> <StaticModInt<M>     > ~ <StaticModInt<M>     > -> StaticModInt<M>  { { |x| x  } ~ { |x| x  } }
        for<M: Modulus> <StaticModInt<M>     > ~ <&'_ StaticModInt<M> > -> StaticModInt<M>  { { |x| x  } ~ { |&x| x } }
        for<M: Modulus> <&'_ StaticModInt<M> > ~ <StaticModInt<M>     > -> StaticModInt<M>  { { |&x| x } ~ { |x| x  } }
        for<M: Modulus> <&'_ StaticModInt<M> > ~ <&'_ StaticModInt<M> > -> StaticModInt<M>  { { |&x| x } ~ { |&x| x } }
        for<I: Id     > <DynamicModInt<I>    > ~ <DynamicModInt<I>    > -> DynamicModInt<I> { { |x| x  } ~ { |x| x  } }
        for<I: Id     > <DynamicModInt<I>    > ~ <&'_ DynamicModInt<I>> -> DynamicModInt<I> { { |x| x  } ~ { |&x| x } }
        for<I: Id     > <&'_ DynamicModInt<I>> ~ <DynamicModInt<I>    > -> DynamicModInt<I> { { |&x| x } ~ { |x| x  } }
        for<I: Id     > <&'_ DynamicModInt<I>> ~ <&'_ DynamicModInt<I>> -> DynamicModInt<I> { { |&x| x } ~ { |&x| x } }

        for<M: Modulus, T: RemEuclidU32> <StaticModInt<M>     > ~ <T> -> StaticModInt<M>  { { |x| x  } ~ { StaticModInt::<M>::new } }
        for<I: Id     , T: RemEuclidU32> <DynamicModInt<I>    > ~ <T> -> DynamicModInt<I> { { |x| x  } ~ { DynamicModInt::<I>::new } }
    }

    impl_assign_ops! {
        for<M: Modulus> <StaticModInt<M> > ~= <StaticModInt<M>     > { _ ~= { |x| x  } }
        for<M: Modulus> <StaticModInt<M> > ~= <&'_ StaticModInt<M> > { _ ~= { |&x| x } }
        for<I: Id     > <DynamicModInt<I>> ~= <DynamicModInt<I>    > { _ ~= { |x| x  } }
        for<I: Id     > <DynamicModInt<I>> ~= <&'_ DynamicModInt<I>> { _ ~= { |&x| x } }

        for<M: Modulus, T: RemEuclidU32> <StaticModInt<M> > ~= <T> { _ ~= { StaticModInt::<M>::new } }
        for<I: Id,      T: RemEuclidU32> <DynamicModInt<I>> ~= <T> { _ ~= { DynamicModInt::<I>::new } }
    }

    macro_rules! impl_folding {
    () => {};
    (impl<$generic_param:ident : $generic_param_bound:tt> $trait:ident<_> for $self:ty { fn $method:ident(_) -> _ { _($unit:expr, $op:expr) } } $($rest:tt)*) => {
        impl<$generic_param: $generic_param_bound> $trait<Self> for $self {
            #[inline]
            fn $method<S>(iter: S) -> Self
            where
                S: Iterator<Item = Self>,
            {
                iter.fold($unit, $op)
            }
        }

        impl<'a, $generic_param: $generic_param_bound> $trait<&'a Self> for $self {
            #[inline]
            fn $method<S>(iter: S) -> Self
            where
                S: Iterator<Item = &'a Self>,
            {
                iter.fold($unit, $op)
            }
        }

        impl_folding!($($rest)*);
    };
}

    impl_folding! {
        impl<M: Modulus> Sum<_>     for StaticModInt<M>  { fn sum(_)     -> _ { _(Self::raw(0), Add::add) } }
        impl<M: Modulus> Product<_> for StaticModInt<M>  { fn product(_) -> _ { _(Self::raw(u32::from(Self::modulus() > 1)), Mul::mul) } }
        impl<I: Id     > Sum<_>     for DynamicModInt<I> { fn sum(_)     -> _ { _(Self::raw(0), Add::add) } }
        impl<I: Id     > Product<_> for DynamicModInt<I> { fn product(_) -> _ { _(Self::raw(u32::from(Self::modulus() > 1)), Mul::mul) } }
    }
}
pub mod ordfloat {
    #![warn(missing_docs)]

    use std::cmp::Ordering;
    use std::hash::{Hash, Hasher};
    use std::ops::Deref;

    #[derive(Copy, Clone, Debug, Default)]
    pub struct OrdFloat(pub f64);

    impl OrdFloat {
        #[must_use]
        pub const fn into_inner(self) -> f64 {
            self.0
        }
    }
    impl From<f64> for OrdFloat {
        fn from(value: f64) -> Self {
            Self(value)
        }
    }
    impl PartialEq for OrdFloat {
        fn eq(&self, other: &Self) -> bool {
            self.cmp(other).is_eq()
        }
    }
    impl Eq for OrdFloat {}
    impl Ord for OrdFloat {
        fn cmp(&self, other: &Self) -> Ordering {
            self.0.total_cmp(&other.0)
        }
    }
    impl PartialOrd for OrdFloat {
        fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
            Some(self.cmp(other))
        }
    }
    impl Hash for OrdFloat {
        fn hash<H: Hasher>(&self, state: &mut H) {
            self.0.to_bits().hash(state);
        }
    }
    impl Deref for OrdFloat {
        type Target = f64;
        fn deref(&self) -> &Self::Target {
            &self.0
        }
    }
}
pub mod pcg64 {
    #![warn(missing_docs)]

    use crate::SmallRng;
    use std::time::{SystemTime, SystemTimeError, UNIX_EPOCH};

    const MULTIPLIER: u128 = 47_026_247_687_942_121_848_144_207_491_837_523_525;
    const DEFAULT_STREAM: u128 = 1_442_695_040_888_963_407;
    const DEFAULT_SEED: u128 = 88_172_645_463_325_252;

    #[derive(Clone, Debug)]
    pub struct PCG64 {
        seed: u128,
        state: u128,
        increment: u128,
    }

    impl Default for PCG64 {
        fn default() -> Self {
            Self::seed_from_u128(DEFAULT_SEED)
        }
    }

    impl PCG64 {
        #[must_use]
        pub fn seed_from_u128(seed: u128) -> Self {
            Self::with_stream(seed, DEFAULT_STREAM)
        }

        #[must_use]
        pub fn with_stream(seed: u128, stream: u128) -> Self {
            let mut rng = Self {
                seed,
                state: 0,
                increment: (stream << 1) | 1,
            };
            let _ = rng.next_u64();
            rng.state = rng.state.wrapping_add(seed);
            let _ = rng.next_u64();
            rng
        }

        pub fn from_system_time() -> Result<Self, SystemTimeError> {
            Ok(Self::seed_from_u128(
                SystemTime::now().duration_since(UNIX_EPOCH)?.as_nanos(),
            ))
        }

        #[must_use]
        pub fn seed(&self) -> u128 {
            self.seed
        }
    }

    impl SmallRng for PCG64 {
        fn next_u64(&mut self) -> u64 {
            self.state = self
                .state
                .wrapping_mul(MULTIPLIER)
                .wrapping_add(self.increment);
            (((self.state >> 64) ^ self.state) as u64).rotate_right((self.state >> 122) as u32)
        }
    }
}
pub mod primenumber {
    #![warn(missing_docs)]

    use std::convert::TryFrom;
    use std::iter::Iterator;

    type ValueType = u64;

    pub trait GetDivisor {
        fn get_divisor(&self) -> Divisor;
    }

    macro_rules! GetDivisor_macro{
    ($($t:ty),*) => {
        $(
        impl GetDivisor for $t {
            fn get_divisor(&self) -> Divisor {
                Divisor::calc(
                    ValueType::try_from(*self)
                        .expect("the value must be a non-negative u64"),
                )
            }
        })*
    };

}

    GetDivisor_macro!(u32, u64, u128, usize, i32, i64, i128, isize);

    pub trait GetPrimeFactorization {
        fn prime_factorization(&self) -> PrimeFactorization<'_>;
    }

    macro_rules! PrimeFactorization_macro{
    ($($t:ty),*) => {
        $(
        impl GetPrimeFactorization for $t {
            fn prime_factorization(&self) -> PrimeFactorization<'_> {
                PrimeFactorization::calc(
                    ValueType::try_from(*self)
                        .expect("the value must be a non-negative u64"),
                )
            }
        })*
    };
}

    PrimeFactorization_macro!(u32, u64, u128, usize, i32, i64, i128, isize);
    pub struct Divisor {
        n: ValueType,
        cur: ValueType,
        flag: bool,
    }

    impl Divisor {
        pub fn calc(n: ValueType) -> Divisor {
            Divisor {
                n,
                cur: 1,
                flag: false,
            }
        }
    }
    impl Iterator for Divisor {
        type Item = ValueType;
        fn next(&mut self) -> Option<Self::Item> {
            if self.cur > self.n / self.cur {
                None
            } else if self.flag {
                if self.cur == self.n / self.cur {
                    return None;
                }
                self.flag = false;
                self.cur += 1;
                Some(self.n / (self.cur - 1))
            } else {
                while !self.n.is_multiple_of(self.cur) {
                    self.cur += 1;
                    if self.cur > self.n / self.cur {
                        return None;
                    }
                }
                self.flag = true;
                Some(self.cur)
            }
        }
    }

    pub struct PrimeFactorization<'a> {
        n: ValueType,
        cur: ValueType,
        p_list: Option<&'a [ValueType]>,
        idx: usize,
    }

    impl<'a> PrimeFactorization<'a> {
        pub fn calc(n: ValueType) -> PrimeFactorization<'a> {
            PrimeFactorization {
                n,
                cur: 1,
                p_list: None,
                idx: 0,
            }
        }
        pub fn calc_fast(n: ValueType, p_list: &'a [ValueType]) -> PrimeFactorization<'a> {
            PrimeFactorization {
                n,
                cur: 1,
                p_list: Some(p_list),
                idx: 0,
            }
        }
    }

    impl Iterator for PrimeFactorization<'_> {
        type Item = ValueType;
        fn next(&mut self) -> Option<Self::Item> {
            loop {
                if self.cur == 0 || self.cur > self.n {
                    return None;
                }

                if let Some(p_list) = self.p_list {
                    if self.idx >= p_list.len() {
                        self.p_list = None;
                        self.cur = if self.cur < 2 {
                            2
                        } else if self.cur == 2 {
                            3
                        } else if self.cur.is_multiple_of(2) {
                            self.cur.checked_add(1)?
                        } else {
                            self.cur.checked_add(2)?
                        };
                    } else {
                        self.cur = p_list[self.idx];
                        self.idx += 1;
                    }
                } else {
                    self.cur = if self.cur < 2 {
                        2
                    } else if self.cur == 2 {
                        3
                    } else {
                        self.cur.checked_add(2)?
                    };
                }

                if self.cur == 0 {
                    return None;
                }
                if self.cur > self.n / self.cur {
                    if self.n != 1 {
                        self.cur = 0;
                        return Some(self.n);
                    }
                    return None;
                }
                if self.n.is_multiple_of(self.cur) {
                    self.n /= self.cur;
                    if self.p_list.is_some() {
                        self.idx -= 1;
                        return Some(self.cur);
                    }
                    let factor = self.cur;
                    self.cur = if factor <= 3 { 1 } else { factor - 2 };
                    return Some(factor);
                }
            }
        }
    }

    pub fn get_primelist(u: ValueType) -> Vec<ValueType> {
        let u = usize::try_from(u).expect("the upper bound must fit in usize");
        let len = u.checked_add(1).expect("the upper bound is too large");
        let mut composite = vec![false; len];
        let mut primes = Vec::new();
        for value in 2..=u {
            if !composite[value] {
                primes.push(value);
            }
            for &prime in &primes {
                if prime > u / value {
                    break;
                }
                composite[value * prime] = true;
                if value.is_multiple_of(prime) {
                    break;
                }
            }
        }
        primes.into_iter().map(|value| value as u64).collect()
    }

    pub fn get_mobius(n: ValueType) -> Vec<isize> {
        let n_usize = usize::try_from(n).expect("the upper bound must fit in usize");
        let len = n_usize
            .checked_add(1)
            .expect("the upper bound is too large");
        let mut mobius = vec![0; len];
        if n_usize == 0 {
            return mobius;
        }
        mobius[1] = 1;
        let mut primes = Vec::new();
        let mut composite = vec![false; len];
        for value in 2..=n_usize {
            if !composite[value] {
                primes.push(value);
                mobius[value] = -1;
            }
            for &prime in &primes {
                if prime > n_usize / value {
                    break;
                }
                let product = value * prime;
                composite[product] = true;
                if value.is_multiple_of(prime) {
                    mobius[product] = 0;
                    break;
                }
                mobius[product] = -mobius[value];
            }
        }
        mobius
    }

    fn modpow_128bit(mut s: u128, mut n: u128, p: u128) -> u128 {
        if p == 0 {
            return 1;
        }
        let mut t = s;
        s = 1;
        while n > 0 {
            if n & 1 != 0 {
                s *= t;
                s %= p;
            }
            n >>= 1;
            t *= t;
            t %= p;
        }
        s
    }

    pub fn miller_rabin(n: u64) -> bool {
        if n == 2 {
            return true;
        }
        if n == 1 || n.is_multiple_of(2) {
            return false;
        }

        let (mut s, mut t) = (0, n - 1);

        while t % 2 == 0 {
            s += 1;
            t >>= 1;
        }

        let bases: &[u64] = if n < 4_759_123_141 {
            &[2, 7, 61]
        } else if n < 341_550_071_728_321 {
            &[2, 3, 5, 7, 11, 13, 17]
        } else if n < 3_825_123_056_546_413_051 {
            &[2, 3, 5, 7, 11, 13, 17, 19, 23]
        } else {
            &[2, 325, 9_375, 28_178, 450_775, 9_780_504, 1_795_265_022]
        };

        'witness: for &base in bases {
            if base >= n {
                continue;
            }
            let mut value = modpow_128bit(base as u128, t as u128, n as u128) as u64;
            if value == 1 || value == n - 1 {
                continue;
            }
            for _ in 1..s {
                value = (u128::from(value) * u128::from(value) % u128::from(n)) as u64;
                if value == n - 1 {
                    continue 'witness;
                }
            }
            return false;
        }
        true
    }

    fn gcd_u64(a: u64, b: u64) -> u64
where
    {
        if b == 0 {
            return a;
        }
        gcd_u64(b, a % b)
    }

    pub struct PollardRho {
        arr: Vec<u64>,
    }

    impl PollardRho {
        pub fn calc(n: u64) -> PollardRho {
            PollardRho { arr: vec![n] }
        }
    }

    fn brent_factor(n: u64) -> u64 {
        if n.is_multiple_of(2) {
            return 2;
        }
        const BATCH: u64 = 128;
        for constant in 1_u64.. {
            let step = |x: u64| {
                ((u128::from(x) * u128::from(x) + u128::from(constant)) % u128::from(n)) as u64
            };
            let (mut y, mut cycle, mut divisor) = (2_u64, 1_u64, 1_u64);
            let (mut x, mut saved_y) = (0_u64, 0_u64);
            while divisor == 1 {
                x = y;
                for _ in 0..cycle {
                    y = step(y);
                }
                let mut offset = 0;
                while offset < cycle && divisor == 1 {
                    saved_y = y;
                    let mut product = 1_u64;
                    for _ in 0..BATCH.min(cycle - offset) {
                        y = step(y);
                        product = ((u128::from(product) * u128::from(x.abs_diff(y)))
                            % u128::from(n)) as u64;
                    }
                    divisor = gcd_u64(product, n);
                    offset += BATCH;
                }
                cycle = cycle.saturating_mul(2);
            }
            if divisor == n {
                loop {
                    saved_y = step(saved_y);
                    divisor = gcd_u64(x.abs_diff(saved_y), n);
                    if divisor > 1 {
                        break;
                    }
                }
            }
            if divisor != n {
                return divisor;
            }
        }
        unreachable!()
    }

    impl Iterator for PollardRho {
        type Item = ValueType;

        fn next(&mut self) -> Option<Self::Item> {
            if self.arr.is_empty() || self.arr[0] == 0 {
                return None;
            }
            let n = self.arr.pop().unwrap();
            if n == 1 {
                return None;
            }
            if miller_rabin(n) {
                let r = n;
                return Some(r);
            }
            if n.is_multiple_of(2) {
                self.arr.push(n / 2);
                return Some(2);
            }

            let divisor = brent_factor(n);
            self.arr.push(n / divisor);
            self.arr.push(divisor);
            self.next()
        }
    }
}
pub mod raq_bit {
    #![warn(missing_docs)]

    use crate::FenwickTree;
    use std::convert::TryFrom;
    use std::ops::{Bound, RangeBounds};

    #[derive(Clone)]
    pub struct RaqBit {
        n: usize,
        bit0: FenwickTree<i128>,
        bit1: FenwickTree<i128>,
    }
    impl RaqBit {
        #[must_use]
        pub fn new(n: usize) -> Self {
            Self {
                n,
                bit0: FenwickTree::new(n, 0),
                bit1: FenwickTree::new(n, 0),
            }
        }
        #[must_use]
        pub fn len(&self) -> usize {
            self.n
        }
        #[must_use]
        pub fn is_empty(&self) -> bool {
            self.n == 0
        }
        pub fn prefix_sum(&self, end: usize) -> i64 {
            i64::try_from(self.prefix_sum_i128(end)).expect("range sum overflow")
        }
        fn prefix_sum_i128(&self, end: usize) -> i128 {
            assert!(end <= self.n);
            let index = i128::try_from(end).expect("index exceeds i128");
            self.bit0
                .accum(end)
                .checked_add(
                    self.bit1
                        .accum(end)
                        .checked_mul(index)
                        .expect("range sum overflow"),
                )
                .expect("range sum overflow")
        }
        pub fn add<R: RangeBounds<usize>>(&mut self, range: R, value: i64) {
            let (l, r) = self.bounds(range);
            if l == r {
                return;
            }
            let li = i128::try_from(l).expect("index exceeds i128");
            let value = i128::from(value);
            self.bit0.add(
                l,
                value
                    .checked_mul(li)
                    .and_then(|x| x.checked_neg())
                    .expect("range update overflow"),
            );
            self.bit1.add(l, value);
            if r < self.n {
                let ri = i128::try_from(r).expect("index exceeds i128");
                self.bit0
                    .add(r, value.checked_mul(ri).expect("range update overflow"));
                self.bit1
                    .add(r, value.checked_neg().expect("range update overflow"));
            }
        }
        pub fn sum<R: RangeBounds<usize>>(&self, range: R) -> i64 {
            let (l, r) = self.bounds(range);
            let sum = self
                .prefix_sum_i128(r)
                .checked_sub(self.prefix_sum_i128(l))
                .expect("range sum overflow");
            i64::try_from(sum).expect("range sum overflow")
        }
        pub fn get(&self, index: usize) -> i64 {
            assert!(index < self.n);
            self.sum(index..=index)
        }
        fn bounds<R: RangeBounds<usize>>(&self, range: R) -> (usize, usize) {
            let l = match range.start_bound() {
                Bound::Included(&x) => x,
                Bound::Excluded(&x) => x.checked_add(1).expect("range bound overflow"),
                Bound::Unbounded => 0,
            };
            let r = match range.end_bound() {
                Bound::Included(&x) => x.checked_add(1).expect("range bound overflow"),
                Bound::Excluded(&x) => x,
                Bound::Unbounded => self.n,
            };
            assert!(l <= r && r <= self.n, "range out of bounds");
            (l, r)
        }
    }
}
pub mod rng_trait {
    #![warn(missing_docs)]

    use std::ops::{Bound, RangeBounds};

    pub trait SmallRng {
        fn next_u64(&mut self) -> u64;

        fn next_u8(&mut self) -> u8 {
            self.next_u64() as u8
        }
        fn next_i8(&mut self) -> i8 {
            self.next_u64() as i8
        }
        fn next_u16(&mut self) -> u16 {
            self.next_u64() as u16
        }
        fn next_i16(&mut self) -> i16 {
            self.next_u64() as i16
        }
        fn next_u32(&mut self) -> u32 {
            self.next_u64() as u32
        }
        fn next_i32(&mut self) -> i32 {
            self.next_u64() as i32
        }
        fn next_i64(&mut self) -> i64 {
            self.next_u64() as i64
        }
        fn next_u128(&mut self) -> u128 {
            (u128::from(self.next_u64()) << 64) | u128::from(self.next_u64())
        }
        fn next_i128(&mut self) -> i128 {
            self.next_u128() as i128
        }
        fn next_usize(&mut self) -> usize {
            self.next_u64() as usize
        }
        fn next_isize(&mut self) -> isize {
            self.next_u64() as isize
        }
        fn next_bool(&mut self) -> bool {
            self.next_u64() & 1 != 0
        }

        fn range_u64<R: RangeBounds<u64>>(&mut self, range: R) -> u64 {
            let start = match range.start_bound() {
                Bound::Included(&x) => u128::from(x),
                Bound::Excluded(&x) => u128::from(x) + 1,
                Bound::Unbounded => 0,
            };
            let end = match range.end_bound() {
                Bound::Included(&x) => u128::from(x) + 1,
                Bound::Excluded(&x) => u128::from(x),
                Bound::Unbounded => 1_u128 << 64,
            };
            assert!(start < end, "random range must be non-empty and ordered");
            (start + sample_below(self, end - start)) as u64
        }

        fn range_i64<R: RangeBounds<i64>>(&mut self, range: R) -> i64 {
            const BIAS: i128 = 1_i128 << 63;
            let start = match range.start_bound() {
                Bound::Included(&x) => i128::from(x) + BIAS,
                Bound::Excluded(&x) => i128::from(x) + BIAS + 1,
                Bound::Unbounded => 0,
            };
            let end = match range.end_bound() {
                Bound::Included(&x) => i128::from(x) + BIAS + 1,
                Bound::Excluded(&x) => i128::from(x) + BIAS,
                Bound::Unbounded => 1_i128 << 64,
            };
            assert!(start < end, "random range must be non-empty and ordered");
            (start + sample_below(self, (end - start) as u128) as i128 - BIAS) as i64
        }

        fn range_usize<R: RangeBounds<usize>>(&mut self, range: R) -> usize {
            let start = match range.start_bound() {
                Bound::Included(&x) => u128::from(x as u64),
                Bound::Excluded(&x) => u128::from(x as u64) + 1,
                Bound::Unbounded => 0,
            };
            let end = match range.end_bound() {
                Bound::Included(&x) => u128::from(x as u64) + 1,
                Bound::Excluded(&x) => u128::from(x as u64),
                Bound::Unbounded => 1_u128 << usize::BITS,
            };
            assert!(start < end, "random range must be non-empty and ordered");
            (start + sample_below(self, end - start)) as usize
        }

        fn range_isize<R: RangeBounds<isize>>(&mut self, range: R) -> isize {
            let bias = 1_i128 << (isize::BITS - 1);
            let start = match range.start_bound() {
                Bound::Included(&x) => x as i128 + bias,
                Bound::Excluded(&x) => x as i128 + bias + 1,
                Bound::Unbounded => 0,
            };
            let end = match range.end_bound() {
                Bound::Included(&x) => x as i128 + bias + 1,
                Bound::Excluded(&x) => x as i128 + bias,
                Bound::Unbounded => 1_i128 << isize::BITS,
            };
            assert!(start < end, "random range must be non-empty and ordered");
            (start + sample_below(self, (end - start) as u128) as i128 - bias) as isize
        }
    }

    fn sample_below<R: SmallRng + ?Sized>(rng: &mut R, width: u128) -> u128 {
        debug_assert!((1..=1_u128 << 64).contains(&width));
        if width == 1_u128 << 64 {
            return u128::from(rng.next_u64());
        }
        let width = width as u64;
        let threshold = width.wrapping_neg() % width;
        loop {
            let value = rng.next_u64();
            if value >= threshold {
                return u128::from(value % width);
            }
        }
    }

    pub trait Shuffle {
        fn shuffle<R: SmallRng>(&mut self, rng: &mut R);
    }

    impl<T> Shuffle for [T] {
        fn shuffle<R: SmallRng>(&mut self, rng: &mut R) {
            for i in (1..self.len()).rev() {
                let j = rng.range_usize(0..=i);
                self.swap(i, j);
            }
        }
    }
}
pub mod rollinghash {
    #![warn(missing_docs)]

    use std::error::Error;
    use std::fmt;
    use std::ops::{Bound, RangeBounds};

    pub const ROLLING_HASH_MODULUS: u64 = (1_u64 << 61) - 1;

    #[derive(Copy, Clone, Debug, Eq, PartialEq)]
    pub enum RollingHashError {
        InvalidBase(u64),
        RangeOutOfBounds { endpoint: usize, len: usize },
        ReversedRange { start: usize, end: usize },
        LengthOutOfBounds { length: usize, max_length: usize },
    }

    impl fmt::Display for RollingHashError {
        fn fmt(&self, formatter: &mut fmt::Formatter<'_>) -> fmt::Result {
            match self {
                Self::InvalidBase(base) => write!(
                    formatter,
                    "rolling-hash base {base} is outside 2..{ROLLING_HASH_MODULUS}"
                ),
                Self::RangeOutOfBounds { endpoint, len } => {
                    write!(
                        formatter,
                        "range endpoint {endpoint} exceeds byte length {len}"
                    )
                }
                Self::ReversedRange { start, end } => {
                    write!(formatter, "rolling-hash range {start}..{end} is reversed")
                }
                Self::LengthOutOfBounds { length, max_length } => write!(
                    formatter,
                    "requested length {length} exceeds cached length {max_length}"
                ),
            }
        }
    }

    impl Error for RollingHashError {}

    #[derive(Clone, Debug, Eq, PartialEq)]
    pub struct RollingHash {
        prefix: Vec<u64>,
        powers: Vec<u64>,
        base: u64,
    }

    impl RollingHash {
        pub fn new(input: impl AsRef<[u8]>, base: u64) -> Result<Self, RollingHashError> {
            if !(2..ROLLING_HASH_MODULUS).contains(&base) {
                return Err(RollingHashError::InvalidBase(base));
            }
            let input = input.as_ref();
            let mut prefix = Vec::with_capacity(input.len() + 1);
            let mut powers = Vec::with_capacity(input.len() + 1);
            prefix.push(0);
            powers.push(1);
            for &byte in input {
                prefix.push(add_mod(
                    mul_mod(*prefix.last().unwrap(), base),
                    u64::from(byte),
                ));
                powers.push(mul_mod(*powers.last().unwrap(), base));
            }
            Ok(Self {
                prefix,
                powers,
                base,
            })
        }

        #[must_use]
        pub fn len(&self) -> usize {
            self.prefix.len() - 1
        }

        #[must_use]
        pub fn is_empty(&self) -> bool {
            self.len() == 0
        }

        #[must_use]
        pub const fn base(&self) -> u64 {
            self.base
        }

        pub fn hash<R: RangeBounds<usize>>(&self, range: R) -> Result<u64, RollingHashError> {
            let (start, end) = self.normalize_range(range)?;
            Ok(sub_mod(
                self.prefix[end],
                mul_mod(self.prefix[start], self.powers[end - start]),
            ))
        }

        pub fn concatenate(
            &self,
            left_hash: u64,
            right_hash: u64,
            right_len: usize,
        ) -> Result<u64, RollingHashError> {
            let Some(&power) = self.powers.get(right_len) else {
                return Err(RollingHashError::LengthOutOfBounds {
                    length: right_len,
                    max_length: self.len(),
                });
            };
            Ok(add_mod(mul_mod(left_hash, power), right_hash))
        }

        fn normalize_range<R: RangeBounds<usize>>(
            &self,
            range: R,
        ) -> Result<(usize, usize), RollingHashError> {
            let start = match range.start_bound() {
                Bound::Included(&value) => value,
                Bound::Excluded(&value) => {
                    value
                        .checked_add(1)
                        .ok_or(RollingHashError::RangeOutOfBounds {
                            endpoint: value,
                            len: self.len(),
                        })?
                }
                Bound::Unbounded => 0,
            };
            let end = match range.end_bound() {
                Bound::Included(&value) => {
                    value
                        .checked_add(1)
                        .ok_or(RollingHashError::RangeOutOfBounds {
                            endpoint: value,
                            len: self.len(),
                        })?
                }
                Bound::Excluded(&value) => value,
                Bound::Unbounded => self.len(),
            };
            if start > end {
                return Err(RollingHashError::ReversedRange { start, end });
            }
            if end > self.len() {
                return Err(RollingHashError::RangeOutOfBounds {
                    endpoint: end,
                    len: self.len(),
                });
            }
            Ok((start, end))
        }
    }

    fn mul_mod(left: u64, right: u64) -> u64 {
        reduce(u128::from(left) * u128::from(right))
    }

    fn add_mod(left: u64, right: u64) -> u64 {
        let sum = left + right;
        if sum >= ROLLING_HASH_MODULUS {
            sum - ROLLING_HASH_MODULUS
        } else {
            sum
        }
    }

    fn sub_mod(left: u64, right: u64) -> u64 {
        if left >= right {
            left - right
        } else {
            left + ROLLING_HASH_MODULUS - right
        }
    }

    fn reduce(product: u128) -> u64 {
        let modulus = u128::from(ROLLING_HASH_MODULUS);
        let folded = (product >> 61) + (product & modulus);
        let folded = (folded >> 61) + (folded & modulus);
        let value = folded as u64;
        if value >= ROLLING_HASH_MODULUS {
            value - ROLLING_HASH_MODULUS
        } else {
            value
        }
    }
}
pub mod scc {

    use crate::internal_scc;

    #[derive(Clone, Debug)]
    pub struct SccGraph {
        internal: internal_scc::SccGraph,
    }

    impl SccGraph {
        pub fn new(n: usize) -> Self {
            SccGraph {
                internal: internal_scc::SccGraph::new(n),
            }
        }

        pub fn add_edge(&mut self, from: usize, to: usize) {
            let n = self.internal.num_vertices();
            assert!(from < n);
            assert!(to < n);
            self.internal.add_edge(from, to);
        }

        pub fn scc(&self) -> Vec<Vec<usize>> {
            self.internal.scc()
        }
    }
}
pub mod segtree {

    use crate::internal_bit::ceil_pow2;
    use crate::internal_type_traits::{BoundedAbove, BoundedBelow, One, Zero};
    use std::cmp::{max, min};
    use std::convert::Infallible;
    use std::iter::FromIterator;
    use std::marker::PhantomData;
    use std::ops::{Add, BitAnd, BitOr, BitXor, Bound, Mul, Not, RangeBounds};

    pub trait Monoid {
        type S: Clone;
        fn identity() -> Self::S;
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S;
    }

    pub struct Max<S>(Infallible, PhantomData<fn() -> S>);
    impl<S> Monoid for Max<S>
    where
        S: Copy + Ord + BoundedBelow,
    {
        type S = S;
        fn identity() -> Self::S {
            S::min_value()
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            max(*a, *b)
        }
    }

    pub struct Min<S>(Infallible, PhantomData<fn() -> S>);
    impl<S> Monoid for Min<S>
    where
        S: Copy + Ord + BoundedAbove,
    {
        type S = S;
        fn identity() -> Self::S {
            S::max_value()
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            min(*a, *b)
        }
    }

    pub struct Additive<S>(Infallible, PhantomData<fn() -> S>);
    impl<S> Monoid for Additive<S>
    where
        S: Copy + Add<Output = S> + Zero,
    {
        type S = S;
        fn identity() -> Self::S {
            S::zero()
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            *a + *b
        }
    }

    pub struct Multiplicative<S>(Infallible, PhantomData<fn() -> S>);
    impl<S> Monoid for Multiplicative<S>
    where
        S: Copy + Mul<Output = S> + One,
    {
        type S = S;
        fn identity() -> Self::S {
            S::one()
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            *a * *b
        }
    }

    pub struct BitwiseOr<S>(Infallible, PhantomData<fn() -> S>);
    impl<S> Monoid for BitwiseOr<S>
    where
        S: Copy + BitOr<Output = S> + Zero,
    {
        type S = S;
        fn identity() -> Self::S {
            S::zero()
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            *a | *b
        }
    }

    pub struct BitwiseAnd<S>(Infallible, PhantomData<fn() -> S>);
    impl<S> Monoid for BitwiseAnd<S>
    where
        S: Copy + BitAnd<Output = S> + Not<Output = S> + Zero,
    {
        type S = S;
        fn identity() -> Self::S {
            !S::zero()
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            *a & *b
        }
    }

    pub struct BitwiseXor<S>(Infallible, PhantomData<fn() -> S>);
    impl<S> Monoid for BitwiseXor<S>
    where
        S: Copy + BitXor<Output = S> + Zero,
    {
        type S = S;
        fn identity() -> Self::S {
            S::zero()
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            *a ^ *b
        }
    }

    impl<M: Monoid> Default for Segtree<M> {
        fn default() -> Self {
            Segtree::new(0)
        }
    }
    impl<M: Monoid> Segtree<M> {
        pub fn new(n: usize) -> Segtree<M> {
            vec![M::identity(); n].into()
        }
    }
    impl<M: Monoid> From<Vec<M::S>> for Segtree<M> {
        fn from(v: Vec<M::S>) -> Self {
            let n = v.len();
            let log = ceil_pow2(n as u32) as usize;
            let size = 1 << log;
            let mut d = vec![M::identity(); 2 * size];
            d[size..][..n].clone_from_slice(&v);
            let mut ret = Segtree { n, size, log, d };
            for i in (1..size).rev() {
                ret.update(i);
            }
            ret
        }
    }
    impl<M: Monoid> FromIterator<M::S> for Segtree<M> {
        fn from_iter<T: IntoIterator<Item = M::S>>(iter: T) -> Self {
            let v = iter.into_iter().collect::<Vec<_>>();
            v.into()
        }
    }
    impl<M: Monoid> Segtree<M> {
        pub fn set(&mut self, mut p: usize, x: M::S) {
            assert!(p < self.n);
            p += self.size;
            self.d[p] = x;
            for i in 1..=self.log {
                self.update(p >> i);
            }
        }

        pub fn get(&self, p: usize) -> M::S {
            assert!(p < self.n);
            self.d[p + self.size].clone()
        }

        pub fn get_slice(&self) -> &[M::S] {
            &self.d[self.size..][..self.n]
        }

        pub fn prod<R>(&self, range: R) -> M::S
        where
            R: RangeBounds<usize>,
        {
            if range.start_bound() == Bound::Unbounded && range.end_bound() == Bound::Unbounded {
                return self.all_prod();
            }

            let mut r = match range.end_bound() {
                Bound::Included(r) => r + 1,
                Bound::Excluded(r) => *r,
                Bound::Unbounded => self.n,
            };
            let mut l = match range.start_bound() {
                Bound::Included(l) => *l,
                Bound::Excluded(l) => l + 1,
                Bound::Unbounded => 0,
            };

            assert!(l <= r && r <= self.n);
            let mut sml = M::identity();
            let mut smr = M::identity();
            l += self.size;
            r += self.size;

            while l < r {
                if l & 1 != 0 {
                    sml = M::binary_operation(&sml, &self.d[l]);
                    l += 1;
                }
                if r & 1 != 0 {
                    r -= 1;
                    smr = M::binary_operation(&self.d[r], &smr);
                }
                l >>= 1;
                r >>= 1;
            }

            M::binary_operation(&sml, &smr)
        }

        pub fn all_prod(&self) -> M::S {
            self.d[1].clone()
        }

        pub fn max_right<F>(&self, mut l: usize, f: F) -> usize
        where
            F: Fn(&M::S) -> bool,
        {
            assert!(l <= self.n);
            assert!(f(&M::identity()));
            if l == self.n {
                return self.n;
            }
            l += self.size;
            let mut sm = M::identity();
            while {
                while l.is_multiple_of(2) {
                    l >>= 1;
                }
                if !f(&M::binary_operation(&sm, &self.d[l])) {
                    while l < self.size {
                        l *= 2;
                        let res = M::binary_operation(&sm, &self.d[l]);
                        if f(&res) {
                            sm = res;
                            l += 1;
                        }
                    }
                    return l - self.size;
                }
                sm = M::binary_operation(&sm, &self.d[l]);
                l += 1;
                {
                    let l = l as isize;
                    (l & -l) != l
                }
            } {}
            self.n
        }

        pub fn min_left<F>(&self, mut r: usize, f: F) -> usize
        where
            F: Fn(&M::S) -> bool,
        {
            assert!(r <= self.n);
            assert!(f(&M::identity()));
            if r == 0 {
                return 0;
            }
            r += self.size;
            let mut sm = M::identity();
            while {
                r -= 1;
                while r > 1 && r % 2 == 1 {
                    r >>= 1;
                }
                if !f(&M::binary_operation(&self.d[r], &sm)) {
                    while r < self.size {
                        r = 2 * r + 1;
                        let res = M::binary_operation(&self.d[r], &sm);
                        if f(&res) {
                            sm = res;
                            r -= 1;
                        }
                    }
                    return r + 1 - self.size;
                }
                sm = M::binary_operation(&self.d[r], &sm);
                {
                    let r = r as isize;
                    (r & -r) != r
                }
            } {}
            0
        }

        fn update(&mut self, k: usize) {
            self.d[k] = M::binary_operation(&self.d[2 * k], &self.d[2 * k + 1]);
        }
    }

    #[derive(Clone)]
    pub struct Segtree<M>
    where
        M: Monoid,
    {
        n: usize,
        size: usize,
        log: usize,
        d: Vec<M::S>,
    }
}
pub mod segtree_template {

    use super::lazysegtree::*;
    use super::segtree::*;

    pub struct MaxAdd;
    impl MapMonoid for MaxAdd {
        type M = Max<i64>;
        type F = i64;

        fn identity_map() -> Self::F {
            0
        }

        fn mapping(&f: &i64, &x: &i64) -> i64 {
            f + x
        }

        fn composition(&f: &i64, &g: &i64) -> i64 {
            f + g
        }
    }

    pub struct MinAdd;
    impl MapMonoid for MinAdd {
        type M = Max<i64>;
        type F = i64;

        fn identity_map() -> Self::F {
            0
        }

        fn mapping(&f: &i64, &x: &i64) -> i64 {
            f + x
        }

        fn composition(&f: &i64, &g: &i64) -> i64 {
            f + g
        }
    }

    const INF: i64 = i64::MAX;

    pub struct MaxForLazy;
    impl Monoid for MaxForLazy {
        type S = i64;
        fn identity() -> Self::S {
            INF
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            if *a == INF {
                *b
            } else if *b == INF {
                *a
            } else {
                std::cmp::max(*a, *b)
            }
        }
    }

    pub struct MinForLazy;
    impl Monoid for MinForLazy {
        type S = i64;
        fn identity() -> Self::S {
            INF
        }
        fn binary_operation(a: &Self::S, b: &Self::S) -> Self::S {
            if *a == INF {
                *b
            } else if *b == INF {
                *a
            } else {
                std::cmp::min(*a, *b)
            }
        }
    }

    pub struct MaxUpdate;
    impl MapMonoid for MaxUpdate {
        type M = MaxForLazy;
        type F = i64;

        fn identity_map() -> Self::F {
            INF
        }

        fn mapping(&f: &i64, &x: &i64) -> i64 {
            if f == INF {
                x
            } else {
                f
            }
        }

        fn composition(&f: &i64, &g: &i64) -> i64 {
            if f == INF {
                g
            } else {
                f
            }
        }
    }

    pub struct MinUpdate;
    impl MapMonoid for MinUpdate {
        type M = MinForLazy;
        type F = i64;

        fn identity_map() -> Self::F {
            INF
        }

        fn mapping(&f: &i64, &x: &i64) -> i64 {
            if f == INF {
                x
            } else {
                f
            }
        }

        fn composition(&f: &i64, &g: &i64) -> i64 {
            if f == INF {
                g
            } else {
                f
            }
        }
    }
}
pub mod string {

    #![allow(
        clippy::many_single_char_names,
        clippy::implicit_saturating_sub,
        clippy::bool_to_int_with_if
    )]

    fn sa_naive<T: Ord>(s: &[T]) -> Vec<usize> {
        let n = s.len();
        let mut sa: Vec<usize> = (0..n).collect();
        sa.sort_by(|&(mut l), &(mut r)| {
            if l == r {
                return std::cmp::Ordering::Equal;
            }
            while l < n && r < n {
                if s[l] != s[r] {
                    return s[l].cmp(&s[r]);
                }
                l += 1;
                r += 1;
            }
            if l == n {
                std::cmp::Ordering::Less
            } else {
                std::cmp::Ordering::Greater
            }
        });
        sa
    }

    fn sa_doubling(s: &[i32]) -> Vec<usize> {
        let n = s.len();
        let mut sa: Vec<usize> = (0..n).collect();
        let mut rnk: Vec<i32> = s.to_vec();
        let mut tmp = vec![0; n];
        let mut k = 1;
        while k < n {
            let cmp = |&x: &usize, &y: &usize| {
                if rnk[x] != rnk[y] {
                    return rnk[x].cmp(&rnk[y]);
                }
                let rx = if x + k < n { rnk[x + k] } else { -1 };
                let ry = if y + k < n { rnk[y + k] } else { -1 };
                rx.cmp(&ry)
            };
            sa.sort_by(cmp);
            tmp[sa[0]] = 0;
            for i in 1..n {
                tmp[sa[i]] =
                    tmp[sa[i - 1]] + i32::from(cmp(&sa[i - 1], &sa[i]) == std::cmp::Ordering::Less);
            }
            std::mem::swap(&mut tmp, &mut rnk);
            k *= 2;
        }
        sa
    }

    trait Threshold {
        fn threshold_naive() -> usize;
        fn threshold_doubling() -> usize;
    }

    enum DefaultThreshold {}
    impl Threshold for DefaultThreshold {
        fn threshold_naive() -> usize {
            10
        }
        fn threshold_doubling() -> usize {
            40
        }
    }

    #[allow(clippy::cognitive_complexity)]
    fn sa_is<T: Threshold>(s: &[usize], upper: usize) -> Vec<usize> {
        let n = s.len();
        match n {
            0 => return vec![],
            1 => return vec![0],
            2 => return if s[0] < s[1] { vec![0, 1] } else { vec![1, 0] },
            _ => (),
        }
        if n < T::threshold_naive() {
            return sa_naive(s);
        }
        if n < T::threshold_doubling() {
            let s: Vec<i32> = s.iter().map(|&x| x as i32).collect();
            return sa_doubling(&s);
        }
        let mut sa = vec![0; n];
        let mut ls = vec![false; n];
        for i in (0..n - 1).rev() {
            ls[i] = if s[i] == s[i + 1] {
                ls[i + 1]
            } else {
                s[i] < s[i + 1]
            };
        }
        let mut sum_l = vec![0; upper + 1];
        let mut sum_s = vec![0; upper + 1];
        for i in 0..n {
            if !ls[i] {
                sum_s[s[i]] += 1;
            } else {
                sum_l[s[i] + 1] += 1;
            }
        }
        for i in 0..=upper {
            sum_s[i] += sum_l[i];
            if i < upper {
                sum_l[i + 1] += sum_s[i];
            }
        }

        let induce = |sa: &mut [usize], lms: &[usize]| {
            for elem in sa.iter_mut() {
                *elem = 0;
            }
            let mut buf = sum_s.clone();
            for &d in lms {
                if d == n {
                    continue;
                }
                let old = buf[s[d]];
                buf[s[d]] += 1;
                sa[old] = d + 1;
            }
            buf.copy_from_slice(&sum_l);
            let old = buf[s[n - 1]];
            buf[s[n - 1]] += 1;
            sa[old] = n;
            for i in 0..n {
                let v = sa[i];
                if v >= 2 && !ls[v - 2] {
                    let old = buf[s[v - 2]];
                    buf[s[v - 2]] += 1;
                    sa[old] = v - 1;
                }
            }
            buf.copy_from_slice(&sum_l);
            for i in (0..n).rev() {
                let v = sa[i];
                if v >= 2 && ls[v - 2] {
                    buf[s[v - 2] + 1] -= 1;
                    sa[buf[s[v - 2] + 1]] = v - 1;
                }
            }
        };
        let mut lms_map = vec![0; n + 1];
        let mut m = 0;
        for i in 1..n {
            if !ls[i - 1] && ls[i] {
                lms_map[i] = m + 1;
                m += 1;
            }
        }
        let mut lms = Vec::with_capacity(m);
        for i in 1..n {
            if !ls[i - 1] && ls[i] {
                lms.push(i);
            }
        }
        assert_eq!(lms.len(), m);
        induce(&mut sa, &lms);

        if m > 0 {
            let mut sorted_lms = Vec::with_capacity(m);
            for &v in &sa {
                if lms_map[v - 1] != 0 {
                    sorted_lms.push(v - 1);
                }
            }
            let mut rec_s = vec![0; m];
            let mut rec_upper = 0;
            rec_s[lms_map[sorted_lms[0]] - 1] = 0;
            for i in 1..m {
                let mut l = sorted_lms[i - 1];
                let mut r = sorted_lms[i];
                let end_l = if lms_map[l] < m { lms[lms_map[l]] } else { n };
                let end_r = if lms_map[r] < m { lms[lms_map[r]] } else { n };
                let same = if end_l - l != end_r - r {
                    false
                } else {
                    while l < end_l {
                        if s[l] != s[r] {
                            break;
                        }
                        l += 1;
                        r += 1;
                    }
                    l != n && s[l] == s[r]
                };
                if !same {
                    rec_upper += 1;
                }
                rec_s[lms_map[sorted_lms[i]] - 1] = rec_upper;
            }

            let rec_sa = sa_is::<T>(&rec_s, rec_upper);
            for i in 0..m {
                sorted_lms[i] = lms[rec_sa[i]];
            }
            induce(&mut sa, &mut sorted_lms);
        }
        for elem in sa.iter_mut() {
            *elem -= 1;
        }
        sa
    }

    fn sa_is_i32<T: Threshold>(s: &[i32], upper: i32) -> Vec<usize> {
        let s: Vec<usize> = s.iter().map(|&x| x as usize).collect();
        sa_is::<T>(&s, upper as usize)
    }

    pub fn suffix_array_manual(s: &[i32], upper: i32) -> Vec<usize> {
        assert!(upper >= 0);
        for &elem in s {
            assert!(0 <= elem && elem <= upper);
        }
        sa_is_i32::<DefaultThreshold>(s, upper)
    }

    pub fn suffix_array_arbitrary<T: Ord>(s: &[T]) -> Vec<usize> {
        let n = s.len();
        let mut idx: Vec<usize> = (0..n).collect();
        idx.sort_by_key(|&i| &s[i]);
        let mut s2 = vec![0; n];
        let mut now = 0;
        for i in 0..n {
            if i > 0 && s[idx[i - 1]] != s[idx[i]] {
                now += 1;
            }
            s2[idx[i]] = now;
        }
        sa_is_i32::<DefaultThreshold>(&s2, now)
    }

    pub fn suffix_array(s: &str) -> Vec<usize> {
        let s2: Vec<usize> = s.bytes().map(|x| x as usize).collect();
        sa_is::<DefaultThreshold>(&s2, 255)
    }

    pub fn lcp_array_arbitrary<T: Ord>(s: &[T], sa: &[usize]) -> Vec<usize> {
        let n = s.len();
        assert!(n >= 1);
        let mut rnk = vec![0; n];
        for i in 0..n {
            rnk[sa[i]] = i;
        }
        let mut lcp = vec![0; n - 1];
        let mut h: usize = 0;
        for i in 0..n - 1 {
            h = h.saturating_sub(1);
            if rnk[i] == 0 {
                continue;
            }
            let j = sa[rnk[i] - 1];
            while j + h < n && i + h < n {
                if s[j + h] != s[i + h] {
                    break;
                }
                h += 1;
            }
            lcp[rnk[i] - 1] = h;
        }
        lcp
    }

    pub fn lcp_array(s: &str, sa: &[usize]) -> Vec<usize> {
        let s: &[u8] = s.as_bytes();
        lcp_array_arbitrary(s, sa)
    }

    pub fn z_algorithm_arbitrary<T: Ord>(s: &[T]) -> Vec<usize> {
        let n = s.len();
        if n == 0 {
            return vec![];
        }
        let mut z = vec![0; n];
        z[0] = 0;
        let mut j = 0;
        for i in 1..n {
            let mut k = if j + z[j] <= i {
                0
            } else {
                std::cmp::min(j + z[j] - i, z[i - j])
            };
            while i + k < n && s[k] == s[i + k] {
                k += 1;
            }
            z[i] = k;
            if j + z[j] < i + z[i] {
                j = i;
            }
        }
        z[0] = n;
        z
    }

    pub fn z_algorithm(s: &str) -> Vec<usize> {
        let s: &[u8] = s.as_bytes();
        z_algorithm_arbitrary(s)
    }
}
pub mod timer {
    #![warn(missing_docs)]

    use std::time::{Duration, Instant};

    #[derive(Clone, Debug)]
    pub struct Timer {
        limit: Duration,
        started: Instant,
    }
    impl Timer {
        #[must_use]
        pub fn new(limit: Duration) -> Self {
            Self {
                limit,
                started: Instant::now(),
            }
        }
        #[must_use]
        pub fn from_millis(milliseconds: u64) -> Self {
            Self::new(Duration::from_millis(milliseconds))
        }
        #[must_use]
        pub fn elapsed(&self) -> Duration {
            self.started.elapsed()
        }
        #[must_use]
        pub fn remaining(&self) -> Duration {
            self.limit.saturating_sub(self.elapsed())
        }
        #[must_use]
        pub fn is_within_limit(&self) -> bool {
            self.elapsed() < self.limit
        }
        pub fn reset(&mut self) {
            self.started = Instant::now();
        }
        #[must_use]
        pub fn formatted_elapsed(&self) -> String {
            format_duration(self.elapsed())
        }
    }
    #[must_use]
    pub fn format_duration(duration: Duration) -> String {
        let nanos = duration.as_nanos();
        if nanos < 1_000 {
            format!("{nanos} ns")
        } else if nanos < 1_000_000 {
            format!("{} μs", duration.as_micros())
        } else if nanos < 1_000_000_000 {
            format!("{} ms", duration.as_millis())
        } else {
            format!("{} s", duration.as_secs())
        }
    }
}
pub mod twosat {
    use crate::internal_scc;

    #[derive(Clone, Debug)]
    pub struct TwoSat {
        n: usize,
        scc: internal_scc::SccGraph,
        answer: Vec<bool>,
    }
    impl TwoSat {
        pub fn new(n: usize) -> Self {
            TwoSat {
                n,
                answer: vec![false; n],
                scc: internal_scc::SccGraph::new(2 * n),
            }
        }
        pub fn add_clause(&mut self, i: usize, f: bool, j: usize, g: bool) {
            assert!(i < self.n && j < self.n);
            self.scc.add_edge(2 * i + !f as usize, 2 * j + g as usize);
            self.scc.add_edge(2 * j + !g as usize, 2 * i + f as usize);
        }
        pub fn satisfiable(&mut self) -> bool {
            let id = self.scc.scc_ids().1;
            for i in 0..self.n {
                if id[2 * i] == id[2 * i + 1] {
                    return false;
                }
                self.answer[i] = id[2 * i] < id[2 * i + 1];
            }
            true
        }
        pub fn answer(&self) -> &[bool] {
            &self.answer
        }
    }
}
pub mod vectools {
    #![warn(missing_docs)]

    use std::cmp::Ordering;

    pub trait BSearch<T> {
        fn lower_bound(&self, key: &T) -> usize;
        fn upper_bound(&self, key: &T) -> usize;
        fn equal_range(&self, key: &T) -> std::ops::Range<usize>;
    }

    impl<T: Ord> BSearch<T> for [T] {
        fn lower_bound(&self, key: &T) -> usize {
            self.partition_point(|x| x < key)
        }
        fn upper_bound(&self, key: &T) -> usize {
            self.partition_point(|x| x <= key)
        }
        fn equal_range(&self, key: &T) -> std::ops::Range<usize> {
            self.lower_bound(key)..self.upper_bound(key)
        }
    }

    pub fn next_permutation<T: Ord>(values: &mut [T]) -> bool {
        next_permutation_by(values, Ord::cmp)
    }

    pub fn next_permutation_by<T, F>(values: &mut [T], mut compare: F) -> bool
    where
        F: FnMut(&T, &T) -> Ordering,
    {
        let Some(pivot) = (1..values.len()).rfind(|&i| compare(&values[i - 1], &values[i]).is_lt())
        else {
            return false;
        };
        let pivot = pivot - 1;
        let swap = (pivot + 1..values.len())
            .rfind(|&i| compare(&values[pivot], &values[i]).is_lt())
            .unwrap();
        values.swap(pivot, swap);
        values[pivot + 1..].reverse();
        true
    }

    pub fn prev_permutation<T: Ord>(values: &mut [T]) -> bool {
        prev_permutation_by(values, Ord::cmp)
    }

    pub fn prev_permutation_by<T, F>(values: &mut [T], mut compare: F) -> bool
    where
        F: FnMut(&T, &T) -> Ordering,
    {
        let Some(pivot) = (1..values.len()).rfind(|&i| compare(&values[i - 1], &values[i]).is_gt())
        else {
            return false;
        };
        let pivot = pivot - 1;
        let swap = (pivot + 1..values.len())
            .rfind(|&i| compare(&values[pivot], &values[i]).is_gt())
            .unwrap();
        values.swap(pivot, swap);
        values[pivot + 1..].reverse();
        true
    }

    #[must_use]
    pub fn coordinate_compress<T: Ord + Clone>(values: &[T]) -> (Vec<usize>, Vec<T>) {
        let mut unique = values.to_vec();
        unique.sort_unstable();
        unique.dedup();
        let ranks = values
            .iter()
            .map(|x| unique.binary_search(x).unwrap())
            .collect();
        (ranks, unique)
    }

    #[must_use]
    pub fn uppercase_to_usize(c: char) -> Option<usize> {
        c.is_ascii_uppercase().then(|| (c as u8 - b'A') as usize)
    }
    #[must_use]
    pub fn lowercase_to_usize(c: char) -> Option<usize> {
        c.is_ascii_lowercase().then(|| (c as u8 - b'a') as usize)
    }
    #[must_use]
    pub fn usize_to_uppercase(index: usize) -> Option<char> {
        (index < 26).then(|| (b'A' + index as u8) as char)
    }
    #[must_use]
    pub fn usize_to_lowercase(index: usize) -> Option<char> {
        (index < 26).then(|| (b'a' + index as u8) as char)
    }
}
pub mod xorshift {
    #![warn(missing_docs)]

    use crate::SmallRng;
    use std::time::{SystemTime, SystemTimeError, UNIX_EPOCH};

    const DEFAULT_SEED: u64 = 88_172_645_463_325_252;
    const NON_ZERO_FALLBACK: u64 = 0x9e37_79b9_7f4a_7c15;

    #[derive(Clone, Debug)]
    pub struct XorShift {
        initial_seed: u64,
        state: u64,
    }

    impl Default for XorShift {
        fn default() -> Self {
            Self::seed_from_u64(DEFAULT_SEED)
        }
    }

    impl XorShift {
        #[must_use]
        pub fn seed_from_u64(seed: u64) -> Self {
            Self {
                initial_seed: seed,
                state: if seed == 0 { NON_ZERO_FALLBACK } else { seed },
            }
        }

        pub fn from_system_time() -> Result<Self, SystemTimeError> {
            let duration = SystemTime::now().duration_since(UNIX_EPOCH)?;
            Ok(Self::seed_from_u64(
                duration.as_secs() ^ u64::from(duration.subsec_nanos()).rotate_left(32),
            ))
        }

        #[must_use]
        pub fn seed(&self) -> u64 {
            self.initial_seed
        }
    }

    impl SmallRng for XorShift {
        fn next_u64(&mut self) -> u64 {
            let mut x = self.state;
            x ^= x >> 12;
            x ^= x << 25;
            x ^= x >> 27;
            self.state = x;
            x.wrapping_mul(0x2545_f491_4f6c_dd1d)
        }
    }
}