#![allow(non_snake_case, unused_imports, unused_macros)]
use itertools::Itertools;
use num::Integer;
use proconio::{fastout, input, marker::Usize1};

macro_rules! debug {
    ($($a:expr),* $(,)*) => {
        #[cfg(debug_assertions)]
        eprintln!(concat!($("| ", stringify!($a), "={:?} "),*, "|"), $(&$a),*);
    };
}
macro_rules! ndvec {
    ($v:expr; $n:expr) => {
        vec![$v; $n]
    };
    ($v:expr; $n:expr, $($ns:expr),+) => {
        vec![ndvec![$v; $($ns),+]; $n]
    };
}
macro_rules! yes_no {
    ($e:expr) => {
        if $e {
            println!("Yes");
        } else {
            println!("No");
        }
    };
}

#[fastout]
fn main() {
    input! {
        t: usize,
        nxyab: [(isize, isize, isize, isize, isize); t]
    }
    for (n, mut x, mut y, a, b) in nxyab {
        let mut i = 0;
        while i < n {
            if x + y >= 0 {
                if b > 0 {
                    let c= Integer::div_ceil(&(x + y + 1), &b).min(n - i);
                    i += c;
                    y -= b * c;
                } else {
                    y -= b * (n - i);
                    i = n;
                }
            } else {
                if a > 0 {
                    let c = Integer::div_ceil(&(-x - y), &a).min(n - i);
                    i += c;
                    x += a * c;
                } else {
                    x += a * (n - i);
                    i = n;
                }
            }
        }
        println!("{x} {y}");
    }
}

#[allow(unused)]
fn partition_point<I: num::Integer + Copy>(mut l: I, mut r: I, mut f: impl FnMut(I) -> bool) {
    let one = I::one();
    let two = one + one;
    while l < r {
        let p = (r - l) / two + l;
        if f(p) {
            l = p + one;
        } else {
            r = p;
        }
    }
}

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

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
{
}

/// Class that has additive identity element
pub trait Zero {
    /// The additive identity element
    fn zero() -> Self;
}

/// Class that has multiplicative identity element
pub trait One {
    /// The multiplicative identity element
    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
);

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

#[allow(dead_code)]
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);
    }

    // Do we need mutable version?
    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
        }
    }
}

use std::cmp::{max, min};
use std::convert::Infallible;
use std::iter::FromIterator;
use std::marker::PhantomData;
use std::ops::{Bound, RangeBounds};

// TODO Should I split monoid-related traits to another module?
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>,
    {
        // Trivial optimization
        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,
            // TODO: There are another way of optimizing [0..r)
            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 {
            // do
            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;
            // while
            {
                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 {
            // do
            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);
            // while
            {
                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]);
    }
}

// Maybe we can use this someday
// ```
// for i in 0..=self.log {
//     for j in 0..1 << i {
//         print!("{}\t", self.d[(1 << i) + j]);
//     }
//     println!();
// }
// ```

#[derive(Clone)]
pub struct Segtree<M>
where
    M: Monoid,
{
    // variable name is _n in original library
    n: usize,
    size: usize,
    log: usize,
    d: Vec<M::S>,
}

pub trait MapMonoid {
    type M: Monoid;
    type F: Clone;
    // type S = <Self::M as Monoid>::S;
    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>,
    {
        // Trivial optimization
        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,
            // TODO: There are another way of optimizing [0..r)
            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,
            // TODO: There are another way of optimizing [0..r)
            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 {
            // do
            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;
            //while
            {
                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 {
            // do
            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);
            // while
            {
                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();
    }
}

// TODO is it useful?
use std::fmt::{Debug, Error, Formatter, Write};
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(())
    }
}

mod internal_math {
    // remove this after dependencies has been added
    #![allow(dead_code)]
    use std::{mem::swap, num::Wrapping as W};

    /// # Arguments
    /// * `m` `1 <= m`
    ///
    /// # Returns
    /// x mod m
    /* const */
    pub(crate) fn safe_mod(mut x: i64, m: i64) -> i64 {
        x %= m;
        if x < 0 {
            x += m;
        }
        x
    }

    /// Fast modular by barrett reduction
    /// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
    /// NOTE: reconsider after Ice Lake
    pub(crate) struct Barrett {
        pub(crate) _m: u32,
        pub(crate) im: u64,
    }

    impl Barrett {
        /// # Arguments
        /// * `m` `1 <= m`
        ///   (Note: `m <= 2^31` should also hold, which is undocumented in the original library.
        ///   See the [pull reqeust commment](https://github.com/rust-lang-ja/ac-library-rs/pull/3#discussion_r484661007)
        ///   for more details.)
        pub(crate) fn new(m: u32) -> Barrett {
            Barrett {
                _m: m,
                im: (-1i64 as u64 / m as u64).wrapping_add(1),
            }
        }

        /// # Returns
        /// `m`
        pub(crate) fn umod(&self) -> u32 {
            self._m
        }

        /// # Parameters
        /// * `a` `0 <= a < m`
        /// * `b` `0 <= b < m`
        ///
        /// # Returns
        /// a * b % 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)
        }
    }

    /// Calculates `a * b % m`.
    ///
    /// * `a` `0 <= a < m`
    /// * `b` `0 <= b < m`
    /// * `m` `1 <= m <= 2^31`
    /// * `im` = ceil(2^64 / `m`)
    #[allow(clippy::many_single_char_names)]
    pub(crate) fn mul_mod(a: u32, b: u32, m: u32, im: u64) -> u32 {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        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
    }

    /// # Parameters
    /// * `n` `0 <= n`
    /// * `m` `1 <= m`
    ///
    /// # Returns
    /// `(x ** n) % m`
    /* 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
    }

    /// Reference:
    /// M. Forisek and J. Jancina,
    /// Fast Primality Testing for Integers That Fit into a Machine Word
    ///
    /// # Parameters
    /// * `n` `0 <= n`
    /* 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
    }

    // omitted
    // template <int n> constexpr bool is_prime = is_prime_constexpr(n);

    /// # Parameters
    /// * `b` `1 <= b`
    ///
    /// # Returns
    /// (g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
    /* 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);
        }

        // Contracts:
        // [1] s - m0 * a = 0 (mod b)
        // [2] t - m1 * a = 0 (mod b)
        // [3] s * |m1| + t * |m0| <= b
        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

            // [3]:
            // (s - t * u) * |m1| + t * |m0 - m1 * u|
            // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
            // = s * |m1| + t * |m0| <= b

            swap(&mut s, &mut t);
            swap(&mut m0, &mut m1);
        }
        // by [3]: |m0| <= b/g
        // by g != b: |m0| < b/g
        if m0 < 0 {
            m0 += b / s;
        }
        (s, m0)
    }

    /// Compile time (currently not) primitive root
    /// @param m must be prime
    /// @return primitive root (and minimum in now)
    /* 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;
        }
    }
    // omitted
    // template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

    /// # Arguments
    /// * `n` `n < 2^32`
    /// * `m` `1 <= m < 2^32`
    ///
    /// # Returns
    /// `sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)`
    /* 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;
            }
            // y_max < m * (n + 1)
            // floor(y_max / m) <= n
            n = y_max / m;
            b = y_max % m;
            std::mem::swap(&mut m, &mut a);
        }
        ans
    }
}

use std::{
    cell::RefCell,
    convert::TryInto as _,
    hash::{Hash, Hasher},
    ops::Neg,
    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> {
    /// Returns the modulus, which is [`<M as Modulus>::VALUE`].
    ///
    /// Corresponds to `atcoder::static_modint::mod` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::ModInt1000000007 as Mint;
    ///
    /// assert_eq!(1_000_000_007, Mint::modulus());
    /// ```
    ///
    /// [`<M as Modulus>::VALUE`]: ../trait.Modulus.html#associatedconstant.VALUE
    #[inline(always)]
    pub fn modulus() -> u32 {
        M::VALUE
    }

    /// Creates a new `StaticModInt`.
    ///
    /// Takes [any primitive integer].
    ///
    /// Corresponds to the constructor of `atcoder::static_modint` in the original ACL.
    ///
    /// [any primitive integer]:  ../trait.RemEuclidU32.html
    #[inline]
    pub fn new<T: RemEuclidU32>(val: T) -> Self {
        Self::raw(val.rem_euclid_u32(M::VALUE))
    }

    /// Constructs a `StaticModInt` from a `val < Self::modulus()` without checking it.
    ///
    /// Corresponds to `atcoder::static_modint::raw` in the original ACL.
    ///
    /// # Constraints
    ///
    /// - `val` is less than `Self::modulus()`
    ///
    /// See [`ModIntBase::raw`] for more more details.
    ///
    /// [`ModIntBase::raw`]: ./trait.ModIntBase.html#tymethod.raw
    #[inline]
    pub fn raw(val: u32) -> Self {
        Self {
            val,
            phantom: PhantomData,
        }
    }

    /// Returns the representative.
    ///
    /// Corresponds to `atcoder::static_modint::val` in the original ACL.
    #[inline]
    pub fn val(self) -> u32 {
        self.val
    }

    /// Returns `self` to the power of `n`.
    ///
    /// Corresponds to `atcoder::static_modint::pow` in the original ACL.
    #[inline]
    pub fn pow(self, n: u64) -> Self {
        <Self as ModIntBase>::pow(self, n)
    }

    /// Returns the multiplicative inverse of `self`.
    ///
    /// Corresponds to `atcoder::static_modint::inv` in the original ACL.
    ///
    /// # Panics
    ///
    /// Panics if the multiplicative inverse does not exist.
    #[inline]
    pub fn inv(self) -> Self {
        if M::HINT_VALUE_IS_PRIME {
            if self.val() == 0 {
                panic!("attempt to divide by zero");
            }

            self.pow((M::VALUE - 2).into())
        } else {
            Self::inv_for_non_prime_modulus(self)
        }
    }
}

/// These methods are implemented for the struct.
/// You don't need to `use` `ModIntBase` to call methods of `StaticModInt`.
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()
    }
}

/// Represents a modulus.
///
/// # Example
///
/// ```
/// macro_rules! modulus {
///     ($($name:ident($value:expr, $is_prime:expr)),*) => {
///         $(
///             #[derive(Copy, Clone, Eq, PartialEq)]
///             enum $name {}
///
///             impl ac_library::modint::Modulus for $name {
///                 const VALUE: u32 = $value;
///                 const HINT_VALUE_IS_PRIME: bool = $is_prime;
///
///                 fn butterfly_cache() -> &'static ::std::thread::LocalKey<::std::cell::RefCell<::std::option::Option<ac_library::modint::ButterflyCache<Self>>>> {
///                     thread_local! {
///                         static BUTTERFLY_CACHE: ::std::cell::RefCell<::std::option::Option<ac_library::modint::ButterflyCache<$name>>> = ::std::default::Default::default();
///                     }
///                     &BUTTERFLY_CACHE
///                 }
///             }
///         )*
///     };
/// }
///
/// use ac_library::StaticModInt;
///
/// modulus!(Mod101(101, true), Mod103(103, true));
///
/// type Z101 = StaticModInt<Mod101>;
/// type Z103 = StaticModInt<Mod103>;
///
/// assert_eq!(Z101::new(101), Z101::new(0));
/// assert_eq!(Z103::new(103), Z103::new(0));
/// ```
pub trait Modulus: 'static + Copy + Eq {
    const VALUE: u32;
    const HINT_VALUE_IS_PRIME: bool;

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

/// Represents $1000000007$.
#[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
    }
}

/// Represents $998244353$.
#[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
    }
}

#[allow(unused)]
/// Cache for butterfly operations.
pub struct ButterflyCache<M> {
    pub(crate) sum_e: Vec<StaticModInt<M>>,
    pub(crate) sum_ie: Vec<StaticModInt<M>>,
}

/// Represents $\mathbb{Z}/m\mathbb{Z}$ where $m$ is a dynamic value.
///
/// Corresponds to `atcoder::dynamic_modint` in the original ACL.
///
/// # Example
///
/// ```
/// use ac_library::ModInt as Mint;
/// use proconio::{input, source::once::OnceSource};
///
/// input! {
///     from OnceSource::from("3 3 7\n"),
///     a: u32,
///     b: u32,
///     m: u32,
/// }
///
/// Mint::set_modulus(m);
/// let a = Mint::new(a);
/// let b = Mint::new(b);
///
/// println!("{}", a * b); // `2`
/// ```
#[derive(Copy, Clone, Eq, PartialEq)]
#[repr(transparent)]
pub struct DynamicModInt<I> {
    val: u32,
    phantom: PhantomData<fn() -> I>,
}

impl<I: Id> DynamicModInt<I> {
    /// Returns the modulus.
    ///
    /// Corresponds to `atcoder::dynamic_modint::mod` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::ModInt as Mint;
    ///
    /// assert_eq!(998_244_353, Mint::modulus()); // default modulus
    /// ```
    #[inline]
    pub fn modulus() -> u32 {
        I::companion_barrett().umod()
    }

    /// Sets a modulus.
    ///
    /// Corresponds to `atcoder::dynamic_modint::set_mod` in the original ACL.
    ///
    /// # Constraints
    ///
    /// - This function must be called earlier than any other operation of `Self`.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::ModInt as Mint;
    ///
    /// Mint::set_modulus(7);
    /// assert_eq!(7, Mint::modulus());
    /// ```
    #[inline]
    pub fn set_modulus(modulus: u32) {
        if modulus == 0 {
            panic!("the modulus must not be 0");
        }
        I::companion_barrett().update(modulus);
    }

    /// Creates a new `DynamicModInt`.
    ///
    /// Takes [any primitive integer].
    ///
    /// Corresponds to the constructor of `atcoder::dynamic_modint` in the original ACL.
    ///
    /// [any primitive integer]:  ../trait.RemEuclidU32.html
    #[inline]
    pub fn new<T: RemEuclidU32>(val: T) -> Self {
        <Self as ModIntBase>::new(val)
    }

    /// Constructs a `DynamicModInt` from a `val < Self::modulus()` without checking it.
    ///
    /// Corresponds to `atcoder::dynamic_modint::raw` in the original ACL.
    ///
    /// # Constraints
    ///
    /// - `val` is less than `Self::modulus()`
    ///
    /// See [`ModIntBase::raw`] for more more details.
    ///
    /// [`ModIntBase::raw`]: ./trait.ModIntBase.html#tymethod.raw
    #[inline]
    pub fn raw(val: u32) -> Self {
        Self {
            val,
            phantom: PhantomData,
        }
    }

    /// Returns the representative.
    ///
    /// Corresponds to `atcoder::static_modint::val` in the original ACL.
    #[inline]
    pub fn val(self) -> u32 {
        self.val
    }

    /// Returns `self` to the power of `n`.
    ///
    /// Corresponds to `atcoder::dynamic_modint::pow` in the original ACL.
    #[inline]
    pub fn pow(self, n: u64) -> Self {
        <Self as ModIntBase>::pow(self, n)
    }

    /// Returns the multiplicative inverse of `self`.
    ///
    /// Corresponds to `atcoder::dynamic_modint::inv` in the original ACL.
    ///
    /// # Panics
    ///
    /// Panics if the multiplicative inverse does not exist.
    #[inline]
    pub fn inv(self) -> Self {
        Self::inv_for_non_prime_modulus(self)
    }
}

/// These methods are implemented for the struct.
/// You don't need to `use` `ModIntBase` to call methods of `DynamicModInt`.
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
    }
}

/// Pair of $m$ and $\lceil 2^{64}/m \rceil$.
pub struct Barrett {
    m: AtomicU32,
    im: AtomicU64,
}

impl Barrett {
    /// Creates a new `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()
    }
}

/// A trait for [`StaticModInt`] and [`DynamicModInt`].
///
/// Corresponds to `atcoder::internal::modint_base` in the original ACL.
///
/// [`StaticModInt`]: ../struct.StaticModInt.html
/// [`DynamicModInt`]: ../struct.DynamicModInt.html
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
{
    /// Returns the modulus.
    ///
    /// Corresponds to `atcoder::static_modint::mod` and `atcoder::dynamic_modint::mod` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::modint::ModIntBase;
    ///
    /// fn f<Z: ModIntBase>() {
    ///     let _: u32 = Z::modulus();
    /// }
    /// ```
    fn modulus() -> u32;

    /// Constructs a `Self` from a `val < Self::modulus()` without checking it.
    ///
    /// Corresponds to `atcoder::static_modint::raw` and `atcoder::dynamic_modint::raw` in the original ACL.
    ///
    /// # Constraints
    ///
    /// - `val` is less than `Self::modulus()`
    ///
    /// **Note that all operations assume that inner values are smaller than the modulus.**
    /// If `val` is greater than or equal to `Self::modulus()`, the behaviors are not defined.
    ///
    /// ```should_panic
    /// use ac_library::ModInt1000000007 as Mint;
    ///
    /// let x = Mint::raw(1_000_000_007);
    /// let y = x + x;
    /// assert_eq!(0, y.val());
    /// ```
    ///
    /// ```text
    /// thread 'main' panicked at 'assertion failed: `(left == right)`
    ///   left: `0`,
    ///  right: `1000000007`', src/modint.rs:8:1
    /// note: run with `RUST_BACKTRACE=1` environment variable to display a backtrace
    /// ```
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::modint::ModIntBase;
    ///
    /// fn f<Z: ModIntBase>() -> Z {
    ///     debug_assert!(Z::modulus() >= 100);
    ///
    ///     let mut acc = Z::new(0);
    ///     for i in 0..100 {
    ///         if i % 3 == 0 {
    ///             // I know `i` is smaller than the modulus!
    ///             acc += Z::raw(i);
    ///         }
    ///     }
    ///     acc
    /// }
    /// ```
    fn raw(val: u32) -> Self;

    /// Returns the representative.
    ///
    /// Corresponds to `atcoder::static_modint::val` and `atcoder::dynamic_modint::val` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::modint::ModIntBase;
    ///
    /// fn f<Z: ModIntBase>(x: Z) {
    ///     let _: u32 = x.val();
    /// }
    /// ```
    fn val(self) -> u32;

    /// Returns the multiplicative inverse of `self`.
    ///
    /// Corresponds to `atcoder::static_modint::inv` and `atcoder::dynamic_modint::inv` in the original ACL.
    ///
    /// # Panics
    ///
    /// Panics if the multiplicative inverse does not exist.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::modint::ModIntBase;
    ///
    /// fn f<Z: ModIntBase>(x: Z) {
    ///     let _: Z = x.inv();
    /// }
    /// ```
    fn inv(self) -> Self;

    /// Creates a new `Self`.
    ///
    /// Takes [any primitive integer].
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::modint::ModIntBase;
    ///
    /// fn f<Z: ModIntBase>() {
    ///     let _ = Z::new(1u32);
    ///     let _ = Z::new(1usize);
    ///     let _ = Z::new(-1i64);
    /// }
    /// ```
    ///
    /// [any primitive integer]:  ../trait.RemEuclidU32.html
    #[inline]
    fn new<T: RemEuclidU32>(val: T) -> Self {
        Self::raw(val.rem_euclid_u32(Self::modulus()))
    }

    /// Returns `self` to the power of `n`.
    ///
    /// Corresponds to `atcoder::static_modint::pow` and `atcoder::dynamic_modint::pow` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library::modint::ModIntBase;
    ///
    /// fn f<Z: ModIntBase>() {
    ///     let _: Z = Z::new(2).pow(3);
    /// }
    /// ```
    #[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
    }
}

/// A trait for `{StaticModInt, DynamicModInt, ModIntBase}::new`.
pub trait RemEuclidU32 {
    /// Calculates `self` $\bmod$ `modulus` losslessly.
    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) } }
}

/// A Disjoint set union (DSU) with union by size and path compression.
///
/// See: [Zvi Galil and Giuseppe F. Italiano, Data structures and algorithms for disjoint set union problems](https://core.ac.uk/download/pdf/161439519.pdf)
///
/// In the following documentation, let $n$ be the size of the DSU.
///
/// # Example
///
/// ```
/// use ac_library::Dsu;
/// use proconio::{input, source::once::OnceSource};
///
/// input! {
///     from OnceSource::from(
///         "5\n\
///          3\n\
///          0 1\n\
///          2 3\n\
///          3 4\n",
///     ),
///     n: usize,
///     abs: [(usize, usize)],
/// }
///
/// let mut dsu = Dsu::new(n);
/// for (a, b) in abs {
///     dsu.merge(a, b);
/// }
///
/// assert!(dsu.same(0, 1));
/// assert!(!dsu.same(1, 2));
/// assert!(dsu.same(2, 4));
///
/// assert_eq!(
///     dsu.groups()
///         .into_iter()
///         .map(|mut group| {
///             group.sort_unstable();
///             group
///         })
///         .collect::<Vec<_>>(),
///     [&[0, 1][..], &[2, 3, 4][..]],
/// );
/// ```
#[derive(Clone, Debug)]
pub struct Dsu {
    n: usize,
    // root node: -1 * component size
    // otherwise: parent
    parent_or_size: Vec<i32>,
}

impl Dsu {
    /// Creates a new `Dsu`.
    ///
    /// # Constraints
    ///
    /// - $0 \leq n \leq 10^8$
    ///
    /// # Complexity
    ///
    /// - $O(n)$
    pub fn new(size: usize) -> Self {
        Self {
            n: size,
            parent_or_size: vec![-1; size],
        }
    }

    // `\textsc` does not work in KaTeX
    /// Performs the Uɴɪᴏɴ operation.
    ///
    /// # Constraints
    ///
    /// - $0 \leq a < n$
    /// - $0 \leq b < n$
    ///
    /// # Panics
    ///
    /// Panics if the above constraints are not satisfied.
    ///
    /// # Complexity
    ///
    /// - $O(\alpha(n))$ amortized
    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
    }

    /// Returns whether the vertices $a$ and $b$ are in the same connected component.
    ///
    /// # Constraints
    ///
    /// - $0 \leq a < n$
    /// - $0 \leq b < n$
    ///
    /// # Panics
    ///
    /// Panics if the above constraint is not satisfied.
    ///
    /// # Complexity
    ///
    /// - $O(\alpha(n))$ amortized
    pub fn same(&mut self, a: usize, b: usize) -> bool {
        assert!(a < self.n);
        assert!(b < self.n);
        self.leader(a) == self.leader(b)
    }

    /// Performs the Fɪɴᴅ operation.
    ///
    /// # Constraints
    ///
    /// - $0 \leq a < n$
    ///
    /// # Panics
    ///
    /// Panics if the above constraint is not satisfied.
    ///
    /// # Complexity
    ///
    /// - $O(\alpha(n))$ amortized
    pub fn leader(&mut self, a: usize) -> usize {
        assert!(a < self.n);
        self._leader(a)
    }

    /// Returns the size of the connected component that contains the vertex $a$.
    ///
    /// # Constraints
    ///
    /// - $0 \leq a < n$
    ///
    /// # Panics
    ///
    /// Panics if the above constraint is not satisfied.
    ///
    /// # Complexity
    ///
    /// - $O(\alpha(n))$ amortized
    pub fn size(&mut self, a: usize) -> usize {
        assert!(a < self.n);
        let x = self.leader(a);
        -self.parent_or_size[x] as usize
    }

    /// Divides the graph into connected components.
    ///
    /// The result may not be ordered.
    ///
    /// # Complexity
    ///
    /// - $O(n)$
    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>>>()
    }

    fn _leader(&mut self, a: usize) -> usize {
        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
    }
}