Problem:
Score : 600 points

Problem Statement

There are 2N people numbered 1 through 2N.
The height of Person i is h_i.

How many ways are there to make N pairs of people such that the following conditions are satisfied?
Compute the answer modulo 998,244,353.

Each person is contained in exactly one pair.

For each pair, the heights of the two people in the pair are different.

Two ways are considered different if for some p and q, Person p and Person q are paired in one way and not in the other.

Constraints

1 \leq N \leq 50,000

1 \leq h_i \leq 100,000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
h_1
:
h_{2N}

Output

Print the answer.

Sample Input 1

2
1
1
2
3

Sample Output 1

2

There are two ways:

Form the pair (Person 1, Person 3) and the pair (Person 2, Person 4).

Form the pair (Person 1, Person 4) and the pair (Person 2, Person 3).

Sample Input 2

5
30
10
20
40
20
10
10
30
50
60

Sample Output 2

516

Solution in Rust (1.42.0) (judge verdict: Time Limit Exceeded)

#![allow(unused_imports)]
#![allow(unused_macros)]
use itertools::Itertools;
use std::cmp::{max, min};
use std::collections::*;
use std::fmt::Write as _Write;
use std::i64;
use std::io::{stdin, BufWriter, Read, Write};
use std::usize;

#[allow(unused_macros)]
macro_rules! parse {
    ($it: ident ) => {};
    ($it: ident, ) => {};
    ($it: ident, $var:ident : $t:tt $($r:tt)*) => {
        let $var = parse_val!($it, $t);
        parse!($it $($r)*);
    };
    ($it: ident, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = parse_val!($it, $t);
        parse!($it $($r)*);
    };
    ($it: ident, $var:ident $($r:tt)*) => {
        let $var = parse_val!($it, usize);
        parse!($it $($r)*);
    };
}

#[allow(unused_macros)]
macro_rules! parse_val {
    ($it: ident, [$t:tt; $len:expr]) => {
        (0..$len).map(|_| parse_val!($it, $t)).collect::<Vec<_>>();
    };
    ($it: ident, ($($t: tt),*)) => {
        ($(parse_val!($it, $t)),*)
    };
    ($it: ident, u1) => {
        $it.next().unwrap().parse::<usize>().unwrap() -1
    };
    ($it: ident, $t: ty) => {
        $it.next().unwrap().parse::<$t>().unwrap()
    };
}

pub fn get_double_factorials<M: Modulus>(n: usize) -> Vec<StaticModInt<M>> {
    let mut ret = vec![0.into(); n];
    ret[0] = 1.into();
    ret[1] = 1.into();
    for i in 2..n {
        ret[i] = ret[i - 2] * StaticModInt::new(i);
    }
    ret
}

type Mint = ModInt998244353;
type M = Mod998244353;
fn solve(s: &str) {
    let mut it = s.split_whitespace();
    parse!(it, n: usize);
    let mut values = HashMap::<usize, usize>::new();
    for _ in 0..2 * n {
        parse!(it, h: usize);
        *values.entry(h).or_default() += 1usize;
    }
    let cm = CombMod::<M>::new(2 * n + 10);
    let df = get_double_factorials::<M>(2 * n + 10);
    let mut q = BTreeMap::new();
    for (h, n) in values {
        let mut p = vec![];
        for i in 0..=(n / 2) {
            p.push(cm.c(n, 2 * i) * if i > 0 { df[2 * i - 1] } else { 1.into() });
        }
        q.insert((!0 - p.len(), h), p);
    }
    while q.len() > 1 {
        let mut it = q.iter();
        let (&k0, p0) = it.next().unwrap();
        let (&k1, p1) = it.next().unwrap();
        let p = convolution(&p0, &p1);
        q.remove(&k0);
        q.remove(&k1);
        q.insert((p.len(), k0.1), p);
    }
    let (_, p) = q.into_iter().next().unwrap();
    let mut ret = Mint::new(0);
    for (i, v) in p.into_iter().enumerate() {
        let x = v * if i < n {
            df[2 * n - 2 * i - 1]
        } else {
            1.into()
        };
        if i % 2 == 0 {
            ret += x
        } else {
            ret -= x;
        }
    }
    println!("{}", ret);
}

fn main() {
    let mut s = String::new();
    stdin().read_to_string(&mut s).unwrap();
    solve(&s);
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_input() {
        let s = "
        ";
        solve(s);
    }
}
trait ChMinMax {
    fn chmin(&mut self, other: Self);
    fn chmax(&mut self, other: Self);
}
impl<T> ChMinMax for T
where
    T: PartialOrd,
{
    fn chmin(&mut self, other: Self) {
        if *self > other {
            *self = other
        }
    }
    fn chmax(&mut self, other: Self) {
        if *self < other {
            *self = other
        }
    }
}

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

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

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

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

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

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

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

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

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

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

        modulus!(M1, M2, M3);

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

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

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

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

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

                // B = 2^63, -B <= x, r(real value) < B
                // (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
                // r = c1[i] (mod MOD1)
                // focus on MOD1
                // r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
                // r = x,
                //     x - M' + (0 or 2B),
                //     x - 2M' + (0, 2B or 4B),
                //     x - 3M' + (0, 2B, 4B or 6B) (without mod!)
                // (r - x) = 0, (0)
                //           - M' + (0 or 2B), (1)
                //           -2M' + (0 or 2B or 4B), (2)
                //           -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
                // we checked that
                //   ((1) mod MOD1) mod 5 = 2
                //   ((2) mod MOD1) mod 5 = 3
                //   ((3) mod MOD1) mod 5 = 4
                let mut diff = c1 - internal_math::safe_mod(x, M1 as _);
                if diff < 0 {
                    diff += M1 as i64;
                }
                x = x.wrapping_sub(OFFSET[diff.rem_euclid(5) as usize] as _);
                x
            })
            .collect()
    }

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

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

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

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

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

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

    #[cfg(test)]
    mod tests {
        #[test]
        fn ceil_pow2() {
            // https://github.com/atcoder/ac-library/blob/2088c8e2431c3f4d29a2cfabc6529fe0a0586c48/test/unittest/bit_test.cpp
            assert_eq!(0, super::ceil_pow2(0));
            assert_eq!(0, super::ceil_pow2(1));
            assert_eq!(1, super::ceil_pow2(2));
            assert_eq!(2, super::ceil_pow2(3));
            assert_eq!(2, super::ceil_pow2(4));
            assert_eq!(3, super::ceil_pow2(5));
            assert_eq!(3, super::ceil_pow2(6));
            assert_eq!(3, super::ceil_pow2(7));
            assert_eq!(3, super::ceil_pow2(8));
            assert_eq!(4, super::ceil_pow2(9));
            assert_eq!(30, super::ceil_pow2(1 << 30));
            assert_eq!(31, super::ceil_pow2((1 << 30) + 1));

            assert_eq!(32, super::ceil_pow2(u32::max_value()));
        }
    }
}
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 {
            // [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) * (self.im as u128)) >> 64) as u64;
            let mut v = z.wrapping_sub(x.wrapping_mul(self._m as u64)) as u32;
            if self._m <= v {
                v = v.wrapping_add(self._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.
    //!
    //! # Major changes from the original ACL
    //!
    //! - Converted the struct names to PascalCase.
    //! - Renamed `mod` → `modulus`.
    //! - Moduli are `u32`, not `i32`.
    //! - `Id`s are `usize`, not `i32`.
    //! - The default `Id` is `0`, not `-1`.
    //! - 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>>`.

    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,
        thread::LocalKey,
    };

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

    /// Corresponds to `atcoder::static_modint` in the original ACL.
    #[derive(Copy, Clone, Eq, PartialEq)]
    #[repr(transparent)]
    pub struct StaticModInt<M> {
        val: u32,
        phantom: PhantomData<fn() -> M>,
    }

    impl<M: Modulus> StaticModInt<M> {
        /// Corresponds to `atcoder::static_modint::mod` in the original ACL.
        #[inline(always)]
        pub fn modulus() -> u32 {
            M::VALUE
        }

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

        /// Corresponds to `atcoder::static_modint::raw` in the original ACL.
        #[inline]
        pub fn raw(val: u32) -> Self {
            Self {
                val,
                phantom: PhantomData,
            }
        }

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

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

        /// 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)
            }
        }
    }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        #[inline]
        pub fn set_modulus(modulus: u32) {
            if modulus == 0 {
                panic!("the modulus must not be 0");
            }
            I::companion_barrett().with(|bt| *bt.borrow_mut() = Barrett::new(modulus))
        }

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

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

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

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

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

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

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

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

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

    pub trait Id: 'static + Copy + Eq {
        // TODO: Make `internal_math::Barret` `Copy`.
        fn companion_barrett() -> &'static LocalKey<RefCell<Barrett>>;
    }

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

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

    pub struct Barrett(internal_math::Barrett);

    impl Barrett {
        #[inline]
        pub fn new(m: u32) -> Self {
            Self(internal_math::Barrett::new(m))
        }

        #[inline]
        fn umod(&self) -> u32 {
            self.0.umod()
        }

        #[inline]
        fn mul(&self, a: u32, b: u32) -> u32 {
            self.0.mul(a, b)
        }
    }

    impl Default for Barrett {
        #[inline]
        fn default() -> Self {
            Self(internal_math::Barrett::new(998_244_353))
        }
    }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    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 } }
    }

    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]));
        }
    }
}
use convolution::*;
use modint::*;

pub fn get_factorials<M: Modulus>(n: usize) -> Vec<StaticModInt<M>> {
    let mut facts = vec![0.into(); n];
    facts[0] = 1.into();
    for i in 1..n {
        facts[i] = facts[i - 1] * StaticModInt::<M>::new(i);
    }
    facts
}

pub fn get_facinvs<M: Modulus>(n: usize, facts: &Vec<StaticModInt<M>>) -> Vec<StaticModInt<M>> {
    let mut invs = vec![0.into(); n];
    invs[n - 1] = facts[n - 1].inv();
    for i in (0..(n - 1)).rev() {
        invs[i] = invs[i + 1] * StaticModInt::new(i + 1)
    }
    invs
}

pub fn gcd(a: usize, b: usize) -> usize {
    match b {
        0 => a,
        _ => gcd(b, a % b),
    }
}

pub fn prime_factors(x: usize) -> Vec<usize> {
    let mut res = vec![];
    let mut y = x;
    let mut p = 2;
    let upper = (y as f64).sqrt() as usize;
    while y > 1 && p <= upper {
        match y % p {
            0 => {
                res.push(p);
                y /= p;
            }
            _ => {
                p += 1;
            }
        }
    }
    if y > 1 {
        res.push(y)
    }
    return res;
}

pub struct CombMod<M: Modulus> {
    facts: Vec<StaticModInt<M>>,
    invs: Vec<StaticModInt<M>>,
}

impl<M: Modulus> CombMod<M> {
    pub fn new(n: usize) -> Self {
        let facts = get_factorials(n);
        let invs = get_facinvs(n, &facts);
        CombMod {
            facts: facts,
            invs: invs,
        }
    }
    pub fn c(&self, a: usize, b: usize) -> StaticModInt<M> {
        if a < b {
            0.into()
        } else {
            (self.facts[a] * self.invs[b]) * self.invs[a - b]
        }
    }
    pub fn p(&self, a: usize, b: usize) -> StaticModInt<M> {
        if b > a {
            0.into()
        } else {
            self.facts[a] * self.invs[a - b]
        }
    }
}