// 2と5が2つずつ以上、かつ0が入っていない

#[allow(unused_macros)]
macro_rules! multi_vec {
    ( $elem:expr; $num:expr ) => (vec![$elem; $num]);
    ( $elem:expr; $num:expr, $($rest:expr),* ) => (vec![multi_vec![$elem; $($rest),*]; $num]);
}

type Mint = ModInt1000000007;

fn main() {
    input! {
        s: {input::bytes},
    };
    let mut memo = Memo::new(s);
    let ans = memo.solve(0, 0, 0, true, true);
    println!("{}", ans);
}

struct Memo {
    s: Vec<u8>,
    // memo[i][two][five][tight][leftmost]
    memo: Vec<Vec<Vec<Vec<Vec<Option<Mint>>>>>>,
}

impl Memo {
    fn new(s: Vec<u8>) -> Memo {
        let memo = multi_vec![None; s.len() + 1, 3, 3, 2, 2];
        Memo { s, memo }
    }

    fn solve(&mut self, idx: usize, two: u8, five: u8, tight: bool, leftmost: bool) -> Mint {
        if let Some(ret) =
            self.memo[idx][two as usize][five as usize][tight as usize][leftmost as usize]
        {
            return ret;
        }

        // 高速化できそう
        // if !tight && two >= 2 && five >= 2 {
        //     return Mint::new(9).pow((self.s.len() - idx) as u64);
        // }
        if idx == self.s.len() {
            if two >= 2 && five >= 2 {
                return 1.into();
            } else {
                return 0.into();
            }
        }

        let d = self.s[idx] - b'0';
        if tight {
            let mut ret = Mint::new(0);
            for i in 1..d {
                let has_two = count_two(i);
                let has_five = count_five(i);
                ret += self.solve(
                    idx + 1,
                    sat2(two + has_two),
                    sat2(five + has_five),
                    false,
                    false,
                );
            }
            if d != 0 {
                let has_two = count_two(d);
                let has_five = count_five(d);
                ret += self.solve(
                    idx + 1,
                    sat2(two + has_two),
                    sat2(five + has_five),
                    true,
                    false,
                );
            }
            if leftmost {
                // 0を許す
                ret += self.solve(idx + 1, 0, 0, false, true);
            }

            self.memo[idx][two as usize][five as usize][tight as usize][leftmost as usize] =
                Some(ret);
            return ret;
        } else {
            let mut ret = Mint::new(0);
            for i in 1..10 {
                let has_two = count_two(i);
                let has_five = count_five(i);
                ret += self.solve(
                    idx + 1,
                    sat2(two + has_two),
                    sat2(five + has_five),
                    false,
                    false,
                );
            }
            if leftmost {
                // 0を許す
                ret += self.solve(idx + 1, 0, 0, false, true);
            }

            self.memo[idx][two as usize][five as usize][tight as usize][leftmost as usize] =
                Some(ret);
            return ret;
        }
    }
}

fn count_two(n: u8) -> u8 {
    match n {
        2 => 1,
        4 => 2,
        6 => 1,
        8 => 3,
        _ => 0,
    }
}

fn count_five(n: u8) -> u8 {
    match n {
        5 => 1,
        _ => 0,
    }
}

fn sat2(n: u8) -> u8 {
    if n >= 2 {
        2
    } else {
        n
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    fn u64_to_bytes(n: u64) -> Vec<u8> {
        n.to_string().as_bytes().to_owned()
    }

    fn assert_n(n: u64) {
        let mut expected = Mint::new(0);
        'outer: for i in 1..=n {
            let mut two = 0u64;
            let mut five = 0u64;
            for d in i.to_string().as_bytes() {
                let d = d - b'0';
                if d == 0 {
                    continue 'outer;
                }
                two += count_two(d) as u64;
                five += count_five(d) as u64;
            }
            if two >= 2 && five >= 2 {
                expected += 1;
            }
        }

        let mut memo = Memo::new(u64_to_bytes(n));
        assert_eq!(expected, memo.solve(0, 0, 0, true, true), "n is {}", n);
    }

    #[test]
    fn test1000() {
        for n in 1..=1000 {
            assert_n(n);
        }
    }

    #[test]
    fn test123456() {
        let n = 123456;
        assert_n(n);
    }
}

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

pub mod internal_math {
    // remove this after dependencies has been added
    #![allow(dead_code)]
    use std::mem::swap;

    /// # 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..std::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);

    #[cfg(test)]
    mod tests {
        #![allow(clippy::unreadable_literal)]
        #![allow(clippy::cognitive_complexity)]
        use crate::internal_math::{inv_gcd, is_prime, pow_mod, primitive_root, safe_mod, Barrett};
        use std::collections::HashSet;

        #[test]
        fn test_safe_mod() {
            assert_eq!(safe_mod(0, 3), 0);
            assert_eq!(safe_mod(1, 3), 1);
            assert_eq!(safe_mod(2, 3), 2);
            assert_eq!(safe_mod(3, 3), 0);
            assert_eq!(safe_mod(4, 3), 1);
            assert_eq!(safe_mod(5, 3), 2);
            assert_eq!(safe_mod(73, 11), 7);
            assert_eq!(safe_mod(2306249155046129918, 6620319213327), 1374210749525);

            assert_eq!(safe_mod(-1, 3), 2);
            assert_eq!(safe_mod(-2, 3), 1);
            assert_eq!(safe_mod(-3, 3), 0);
            assert_eq!(safe_mod(-4, 3), 2);
            assert_eq!(safe_mod(-5, 3), 1);
            assert_eq!(safe_mod(-7170500492396019511, 777567337), 333221848);
        }

        #[test]
        fn test_barrett() {
            let b = Barrett::new(7);
            assert_eq!(b.umod(), 7);
            assert_eq!(b.mul(2, 3), 6);
            assert_eq!(b.mul(4, 6), 3);
            assert_eq!(b.mul(5, 0), 0);

            let b = Barrett::new(998244353);
            assert_eq!(b.umod(), 998244353);
            assert_eq!(b.mul(2, 3), 6);
            assert_eq!(b.mul(3141592, 653589), 919583920);
            assert_eq!(b.mul(323846264, 338327950), 568012980);

            // make `z - x * self._m as u64` overflow.
            // Thanks @koba-e964 (at https://github.com/rust-lang-ja/ac-library-rs/pull/3#discussion_r484932161)
            let b = Barrett::new(2147483647);
            assert_eq!(b.umod(), 2147483647);
            assert_eq!(b.mul(1073741824, 2147483645), 2147483646);
        }

        #[test]
        fn test_pow_mod() {
            assert_eq!(pow_mod(0, 0, 1), 0);
            assert_eq!(pow_mod(0, 0, 3), 1);
            assert_eq!(pow_mod(0, 0, 723), 1);
            assert_eq!(pow_mod(0, 0, 998244353), 1);
            assert_eq!(pow_mod(0, 0, i32::max_value()), 1);

            assert_eq!(pow_mod(0, 1, 1), 0);
            assert_eq!(pow_mod(0, 1, 3), 0);
            assert_eq!(pow_mod(0, 1, 723), 0);
            assert_eq!(pow_mod(0, 1, 998244353), 0);
            assert_eq!(pow_mod(0, 1, i32::max_value()), 0);

            assert_eq!(pow_mod(0, i64::max_value(), 1), 0);
            assert_eq!(pow_mod(0, i64::max_value(), 3), 0);
            assert_eq!(pow_mod(0, i64::max_value(), 723), 0);
            assert_eq!(pow_mod(0, i64::max_value(), 998244353), 0);
            assert_eq!(pow_mod(0, i64::max_value(), i32::max_value()), 0);

            assert_eq!(pow_mod(1, 0, 1), 0);
            assert_eq!(pow_mod(1, 0, 3), 1);
            assert_eq!(pow_mod(1, 0, 723), 1);
            assert_eq!(pow_mod(1, 0, 998244353), 1);
            assert_eq!(pow_mod(1, 0, i32::max_value()), 1);

            assert_eq!(pow_mod(1, 1, 1), 0);
            assert_eq!(pow_mod(1, 1, 3), 1);
            assert_eq!(pow_mod(1, 1, 723), 1);
            assert_eq!(pow_mod(1, 1, 998244353), 1);
            assert_eq!(pow_mod(1, 1, i32::max_value()), 1);

            assert_eq!(pow_mod(1, i64::max_value(), 1), 0);
            assert_eq!(pow_mod(1, i64::max_value(), 3), 1);
            assert_eq!(pow_mod(1, i64::max_value(), 723), 1);
            assert_eq!(pow_mod(1, i64::max_value(), 998244353), 1);
            assert_eq!(pow_mod(1, i64::max_value(), i32::max_value()), 1);

            assert_eq!(pow_mod(i64::max_value(), 0, 1), 0);
            assert_eq!(pow_mod(i64::max_value(), 0, 3), 1);
            assert_eq!(pow_mod(i64::max_value(), 0, 723), 1);
            assert_eq!(pow_mod(i64::max_value(), 0, 998244353), 1);
            assert_eq!(pow_mod(i64::max_value(), 0, i32::max_value()), 1);

            assert_eq!(pow_mod(i64::max_value(), i64::max_value(), 1), 0);
            assert_eq!(pow_mod(i64::max_value(), i64::max_value(), 3), 1);
            assert_eq!(pow_mod(i64::max_value(), i64::max_value(), 723), 640);
            assert_eq!(
                pow_mod(i64::max_value(), i64::max_value(), 998244353),
                683296792
            );
            assert_eq!(
                pow_mod(i64::max_value(), i64::max_value(), i32::max_value()),
                1
            );

            assert_eq!(pow_mod(2, 3, 1_000_000_007), 8);
            assert_eq!(pow_mod(5, 7, 1_000_000_007), 78125);
            assert_eq!(pow_mod(123, 456, 1_000_000_007), 565291922);
        }

        #[test]
        fn test_is_prime() {
            assert!(!is_prime(0));
            assert!(!is_prime(1));
            assert!(is_prime(2));
            assert!(is_prime(3));
            assert!(!is_prime(4));
            assert!(is_prime(5));
            assert!(!is_prime(6));
            assert!(is_prime(7));
            assert!(!is_prime(8));
            assert!(!is_prime(9));

            // assert!(is_prime(57));
            assert!(!is_prime(57));
            assert!(!is_prime(58));
            assert!(is_prime(59));
            assert!(!is_prime(60));
            assert!(is_prime(61));
            assert!(!is_prime(62));

            assert!(!is_prime(701928443));
            assert!(is_prime(998244353));
            assert!(!is_prime(1_000_000_000));
            assert!(is_prime(1_000_000_007));

            assert!(is_prime(i32::max_value()));
        }

        #[test]
        fn test_is_prime_sieve() {
            let n = 1_000_000;
            let mut prime = vec![true; n];
            prime[0] = false;
            prime[1] = false;
            for i in 0..n {
                assert_eq!(prime[i], is_prime(i as i32));
                if prime[i] {
                    for j in (2 * i..n).step_by(i) {
                        prime[j] = false;
                    }
                }
            }
        }

        #[test]
        fn test_inv_gcd() {
            for &(a, b, g) in &[
                (0, 1, 1),
                (0, 4, 4),
                (0, 7, 7),
                (2, 3, 1),
                (-2, 3, 1),
                (4, 6, 2),
                (-4, 6, 2),
                (13, 23, 1),
                (57, 81, 3),
                (12345, 67890, 15),
                (-3141592 * 6535, 3141592 * 8979, 3141592),
                (i64::max_value(), i64::max_value(), i64::max_value()),
                (i64::min_value(), i64::max_value(), 1),
            ] {
                let (g_, x) = inv_gcd(a, b);
                assert_eq!(g, g_);
                let b_ = b as i128;
                assert_eq!(((x as i128 * a as i128) % b_ + b_) % b_, g as i128 % b_);
            }
        }

        #[test]
        fn test_primitive_root() {
            for &p in &[
                2,
                3,
                5,
                7,
                233,
                200003,
                998244353,
                1_000_000_007,
                i32::max_value(),
            ] {
                assert!(is_prime(p));
                let g = primitive_root(p);
                if p != 2 {
                    assert_ne!(g, 1);
                }

                let q = p - 1;
                for i in (2..i32::max_value()).take_while(|i| i * i <= q) {
                    if q % i != 0 {
                        break;
                    }
                    for &r in &[i, q / i] {
                        assert_ne!(pow_mod(g as i64, r as i64, p), 1);
                    }
                }
                assert_eq!(pow_mod(g as i64, q as i64, p), 1);

                if p < 1_000_000 {
                    assert_eq!(
                        (0..p - 1)
                            .scan(1, |i, _| {
                                *i = *i * g % p;
                                Some(*i)
                            })
                            .collect::<HashSet<_>>()
                            .len() as i32,
                        p - 1
                    );
                }
            }
        }
    }
}
pub mod modint {
    //! Structs that treat the modular arithmetic.
    //!
    //! For most of the problems, It is sufficient to use [`ModInt1000000007`] or [`ModInt998244353`], which can be used as follows.
    //!
    //! ```
    //! use ac_library_rs::ModInt1000000007 as Mint; // rename to whatever you want
    //! use proconio::{input, source::once::OnceSource};
    //!
    //! input! {
    //!     from OnceSource::from("1000000006 2\n"),
    //!     a: Mint,
    //!     b: Mint,
    //! }
    //!
    //! println!("{}", a + b); // `1`
    //! ```
    //!
    //! If the modulus is not fixed, you can use [`ModInt`] as follows.
    //!
    //! ```
    //! use ac_library_rs::ModInt as Mint; // rename to whatever you want
    //! 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`
    //! ```
    //!
    //! # Major changes from the original ACL
    //!
    //! - Converted the struct names to PascalCase.
    //! - Renamed `mod` → `modulus`.
    //! - Moduli are `u32`, not `i32`.
    //! - Each `Id` does not have a identifier number. Instead, they explicitly own `&'static LocalKey<RefCell<Barrett>>`.
    //! - The type of the argument of `pow` is `u64`, not `i64`.
    //! - Modints implement `FromStr` and `Display`. Modints in the original ACL don't have `operator<<` or `operator>>`.
    //!
    //! [`ModInt1000000007`]: ./type.ModInt1000000007.html
    //! [`ModInt998244353`]: ./type.ModInt998244353.html
    //! [`ModInt`]: ./type.ModInt.html

    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>;

    /// Represents _ℤ/mℤ_ where _m_ is a constant value.
    ///
    /// Corresponds to `atcoder::static_modint` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library_rs::ModInt1000000007 as Mint;
    /// use proconio::{input, source::once::OnceSource};
    ///
    /// input! {
    ///     from OnceSource::from("1000000006 2\n"),
    ///     a: Mint,
    ///     b: Mint,
    /// }
    ///
    /// println!("{}", a + b); // `1`
    /// ```
    #[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_rs::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,
            }
        }

        /// Retruns 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)
        }

        /// Retruns 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");
                }
                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)
            }
        }
    }

    /// 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_rs::modint::Modulus for $name {
    ///                 const VALUE: u32 = $value;
    ///                 const HINT_VALUE_IS_PRIME: bool = $is_prime;
    ///
    ///                 fn butterfly_cache() -> &'static ::std::thread::LocalKey<::std::cell::RefCell<::std::option::Option<ac_library_rs::modint::ButterflyCache<Self>>>> {
    ///                     thread_local! {
    ///                         static BUTTERFLY_CACHE: ::std::cell::RefCell<::std::option::Option<ac_library_rs::modint::ButterflyCache<$name>>> = ::std::default::Default::default();
    ///                     }
    ///                     &BUTTERFLY_CACHE
    ///                 }
    ///             }
    ///         )*
    ///     };
    /// }
    ///
    /// use ac_library_rs::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
        }
    }

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

    /// Represents _ℤ/mℤ_ where _m_ is a dynamic value.
    ///
    /// Corresponds to `atcoder::dynamic_modint` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library_rs::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_rs::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_rs::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,
            }
        }

        /// Retruns 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)
        }

        /// Retruns 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 _ceil(2⁶⁴/m)_.
    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_rs::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_rs::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_rs::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;

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

        /// Retruns 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_rs::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_rs::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_rs::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(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` _mod_ `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::derive_hash_xor_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(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(1), Mul::mul) } }
    }

    #[cfg(test)]
    mod tests {
        use crate::modint::ModInt1000000007;

        #[test]
        fn static_modint_new() {
            assert_eq!(0, ModInt1000000007::new(0u32).val);
            assert_eq!(1, ModInt1000000007::new(1u32).val);
            assert_eq!(1, ModInt1000000007::new(1_000_000_008u32).val);

            assert_eq!(0, ModInt1000000007::new(0u64).val);
            assert_eq!(1, ModInt1000000007::new(1u64).val);
            assert_eq!(1, ModInt1000000007::new(1_000_000_008u64).val);

            assert_eq!(0, ModInt1000000007::new(0usize).val);
            assert_eq!(1, ModInt1000000007::new(1usize).val);
            assert_eq!(1, ModInt1000000007::new(1_000_000_008usize).val);

            assert_eq!(0, ModInt1000000007::new(0i64).val);
            assert_eq!(1, ModInt1000000007::new(1i64).val);
            assert_eq!(1, ModInt1000000007::new(1_000_000_008i64).val);
            assert_eq!(1_000_000_006, ModInt1000000007::new(-1i64).val);
        }

        #[test]
        fn static_modint_add() {
            fn add(lhs: u32, rhs: u32) -> u32 {
                (ModInt1000000007::new(lhs) + ModInt1000000007::new(rhs)).val
            }

            assert_eq!(2, add(1, 1));
            assert_eq!(1, add(1_000_000_006, 2));
        }

        #[test]
        fn static_modint_sub() {
            fn sub(lhs: u32, rhs: u32) -> u32 {
                (ModInt1000000007::new(lhs) - ModInt1000000007::new(rhs)).val
            }

            assert_eq!(1, sub(2, 1));
            assert_eq!(1_000_000_006, sub(0, 1));
        }

        #[test]
        fn static_modint_mul() {
            fn mul(lhs: u32, rhs: u32) -> u32 {
                (ModInt1000000007::new(lhs) * ModInt1000000007::new(rhs)).val
            }

            assert_eq!(1, mul(1, 1));
            assert_eq!(4, mul(2, 2));
            assert_eq!(999_999_937, mul(100_000, 100_000));
        }

        #[test]
        fn static_modint_prime_div() {
            fn div(lhs: u32, rhs: u32) -> u32 {
                (ModInt1000000007::new(lhs) / ModInt1000000007::new(rhs)).val
            }

            assert_eq!(0, div(0, 1));
            assert_eq!(1, div(1, 1));
            assert_eq!(1, div(2, 2));
            assert_eq!(23_809_524, div(1, 42));
        }

        #[test]
        fn static_modint_sum() {
            fn sum(values: &[i64]) -> ModInt1000000007 {
                values.iter().copied().map(ModInt1000000007::new).sum()
            }

            assert_eq!(ModInt1000000007::new(-3), sum(&[-1, 2, -3, 4, -5]));
        }

        #[test]
        fn static_modint_product() {
            fn product(values: &[i64]) -> ModInt1000000007 {
                values.iter().copied().map(ModInt1000000007::new).product()
            }

            assert_eq!(ModInt1000000007::new(-120), product(&[-1, 2, -3, 4, -5]));
        }

        #[test]
        fn static_modint_binop_coercion() {
            let f = ModInt1000000007::new;
            let a = 10_293_812_usize;
            let b = 9_083_240_982_usize;
            assert_eq!(f(a) + f(b), f(a) + b);
            assert_eq!(f(a) - f(b), f(a) - b);
            assert_eq!(f(a) * f(b), f(a) * b);
            assert_eq!(f(a) / f(b), f(a) / b);
        }

        #[test]
        fn static_modint_assign_coercion() {
            let f = ModInt1000000007::new;
            let a = f(10_293_812_usize);
            let b = 9_083_240_982_usize;
            let expected = (((a + b) * b) - b) / b;
            let mut c = a;
            c += b;
            c *= b;
            c -= b;
            c /= b;
            assert_eq!(expected, c);
        }
    }
}
use modint::*;

// from https://github.com/qryxip/competitive-programming-library/blob/master/crates/io/sourcefiles/input.rs
mod input {
    //! Provides `input!` macro.
    //!
    //! # Example
    //!
    //! ```no_run
    //! #[macro_use]
    //! extern crate input as _;
    //!
    //! fn main() {
    //!     // https://atcoder.jp/contests/abc166/tasks/abc166_b
    //!
    //!     input! {
    //!         n: usize,
    //!         ass: [[{ input::usize1 }]],
    //!     }
    //!
    //!     let _: usize = n;
    //!     let _: Vec<Vec<usize>> = ass;
    //! }
    //! ```

    use std::{
        cell::RefCell,
        fmt::Debug,
        io::{self, BufRead, Read},
        rc::Rc,
        str::{FromStr, SplitWhitespace},
    };

    #[macro_export]
    macro_rules! input {
        (from $scanner:ident; $($tt:tt)*) => {
            $crate::input_inner!(@scanner($scanner), @tts($($tt)*))
        };
        ($($tt:tt)*) => {
            let __scanner = $crate::input::DEFAULT_SCANNER.with(|__scanner| __scanner.clone());
            let mut __scanner_ref = __scanner.borrow_mut();
            if let $crate::input::Scanner::Uninited = *__scanner_ref {
                *__scanner_ref = $crate::input::Scanner::stdin_auto().unwrap();
            }
            $crate::input_inner!(@scanner(__scanner_ref), @tts($($tt)*));
            ::std::mem::drop(__scanner_ref);
            ::std::mem::drop(__scanner);
        };
    }

    #[macro_export]
    macro_rules! input_inner {
        (@scanner($scanner:ident), @tts()) => {};
        (@scanner($scanner:ident), @tts(mut $single_tt_pat:tt : $readable:tt)) => {
            let mut $single_tt_pat = $crate::read!(from $scanner { $readable });
        };
        (@scanner($scanner:ident), @tts($single_tt_pat:tt : $readable:tt)) => {
            let $single_tt_pat = $crate::read!(from $scanner { $readable });
        };
        (@scanner($scanner:ident), @tts(mut $single_tt_pat:tt : $readable:tt, $($rest:tt)*)) => {
            $crate::input_inner!(@scanner($scanner), @tts(mut $single_tt_pat: $readable));
            $crate::input_inner!(@scanner($scanner), @tts($($rest)*));
        };
        (@scanner($scanner:ident), @tts($single_tt_pat:tt : $readable:tt, $($rest:tt)*)) => {
            $crate::input_inner!(@scanner($scanner), @tts($single_tt_pat: $readable));
            $crate::input_inner!(@scanner($scanner), @tts($($rest)*));
        };
    }

    #[macro_export]
    macro_rules! read {
        (from $scanner:ident { [$tt:tt] }) => {
            $crate::read!(from $scanner { [$tt; $crate::read!(from $scanner { usize })] })
        };
        (from $scanner:ident  { [$tt:tt; $n:expr] }) => {
            (0..$n).map(|_| $crate::read!(from $scanner { $tt })).collect::<Vec<_>>()
        };
        (from $scanner:ident { ($($tt:tt),+) }) => {
            ($($crate::read!(from $scanner { $tt })),*)
        };
        (from $scanner:ident { { $f:expr } }) => {
            $crate::input::FnOnceExt::<_>::call_once_from_reader($f, &mut $scanner)
        };
        (from $scanner:ident { $ty:ty }) => {
            <$ty as $crate::input::Readable>::read_from_scanner(&mut $scanner)
        };
    }

    #[macro_export]
    macro_rules! readable {
    ($name:ident; |$scanner:ident| { $($body:tt)* }) => {
        $crate::readable!($name; |$scanner| -> () { $($body)* });
    };
    ($name:ident; |$scanner:ident| $expr:expr) => {
        $crate::readable!($name; |$scanner| -> () { $expr });
    };
    ($name:ident; |$scanner:ident| -> $output:ty { $($body:tt)* }) => {
        enum $name {}

        impl $crate::input::Readable for $name {
            type Output = $output;

            fn read_from_scanner(mut $scanner: &mut $crate::input::Scanner) -> $output {
                $($body)*
            }
        }
    };
}

    #[inline]
    pub fn usize1(n: usize) -> usize {
        n - 1
    }

    #[inline]
    pub fn bytes(s: String) -> Vec<u8> {
        s.into()
    }

    #[doc(hidden)]
    pub trait FnOnceExt<A> {
        type Output;
        fn call_once_from_reader(this: Self, scanner: &mut Scanner) -> Self::Output;
    }

    impl<A, O, F> FnOnceExt<A> for F
    where
        A: FromStr,
        A::Err: Debug,
        F: FnOnce(A) -> O,
    {
        type Output = O;

        #[inline]
        fn call_once_from_reader(this: Self, scanner: &mut Scanner) -> O {
            this(A::read_from_scanner(scanner))
        }
    }

    pub enum Scanner {
        Uninited,
        Once {
            words: SplitWhitespace<'static>,
        },
        Lines {
            rdr: Box<dyn BufRead>,
            words: SplitWhitespace<'static>,
        },
    }

    impl Scanner {
        pub fn stdin_auto() -> io::Result<Self> {
            if cfg!(debug_assertions) {
                Ok(Self::lines(Box::leak(Box::new(io::stdin())).lock()))
            } else {
                Self::once(io::stdin())
            }
        }

        pub fn once<R: Read>(mut rdr: R) -> io::Result<Self> {
            let mut buf = String::with_capacity(1024);
            rdr.read_to_string(&mut buf)?;
            let words = Box::leak(buf.into_boxed_str()).split_whitespace();
            Ok(Self::Once { words })
        }

        pub fn lines<R: BufRead + 'static>(rdr: R) -> Self {
            Self::Lines {
                rdr: Box::new(rdr),
                words: "".split_whitespace(),
            }
        }

        pub fn parse_next_unwrap<T: FromStr>(&mut self) -> T
        where
            T::Err: Debug,
        {
            match self {
                Self::Uninited => None,
                Self::Once { words } => words.next(),
                Self::Lines { rdr, words } => words.next().or_else(|| {
                    let mut line = "".to_owned();
                    rdr.read_line(&mut line).unwrap();
                    *words = Box::leak(line.into_boxed_str()).split_whitespace();
                    words.next()
                }),
            }
            .expect("reached EOF")
            .parse()
            .unwrap()
        }
    }

    thread_local! {
        pub static DEFAULT_SCANNER: Rc<RefCell<Scanner>> = Rc::new(RefCell::new(Scanner::Uninited));
    }

    pub trait Readable {
        type Output;

        fn read_from_scanner(scanner: &mut Scanner) -> Self::Output;
    }

    impl<T: FromStr> Readable for T
    where
        T::Err: Debug,
    {
        type Output = Self;

        fn read_from_scanner(scanner: &mut Scanner) -> Self {
            scanner.parse_next_unwrap()
        }
    }
}