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)

use factorial::Factorial;
#[allow(unused_imports)]
use ngtio::prelude::*;

use polynomial::Poly;
use type_traits::One;

type Fp = fp::F998244353;

fn g(k: usize, fact: &Factorial<Fp>, pow_2_inv: &[Fp]) -> Poly<Fp> {
    Poly::new(
        (0..)
            .take_while(|&i| 2 * i <= k)
            .map(|i| fact[k] * fact.inv(k - 2 * i) * fact.inv(i) * pow_2_inv[i])
            .collect(),
    )
}

fn main() {
    let mut buf = ngtio::with_stdin();
    let n = buf.usize();
    let fact = Factorial::new(2 * n + 1, |x| Fp::new(x as i64));
    let pow_2_inv = std::iter::successors(Some(Fp::one()), |x| Some(x * Fp::new(2).inv()))
        .take(2 * n + 1)
        .collect::<Vec<_>>();
    let mut double_factorial = vec![Fp::one(); 2 * n + 1];
    for i in 2..=2 * n {
        double_factorial[i] = double_factorial[i - 2] * Fp::new(i as i64);
    }

    let mut m = vec![0; 100_000];
    for _ in 0..2 * n {
        let x = buf.parse::<Usize1>();
        m[x] += 1;
    }
    let mut polies = m
        .iter()
        .copied()
        .filter(|&x| 2 <= x)
        .map(|x| g(x, &fact, &pow_2_inv))
        .collect::<Vec<_>>();
    polies.sort_by_key(|v| v.0.len());
    let mul = polies
        .iter()
        .cloned()
        .fold(Poly::one(), |x, y| fft::multiply(x, y));
    let ans = mul
        .into_inner()
        .iter()
        .enumerate()
        .map(|(i, &x)| {
            let sign = if i % 2 == 0 { Fp::one() } else { -Fp::one() };
            x * sign * double_factorial[(2 * (n - i)).saturating_sub(1)]
        })
        .sum::<Fp>();
    println!("{}", ans);
}

// dbg {{{
#[allow(dead_code)]
mod dbg {
    #[macro_export]
    macro_rules! lg {
        () => {
            $crate::eprintln!("[{}:{}]", $crate::file!(), $crate::line!());
        };
        ($val:expr) => {
            match $val {
                tmp => {
                    eprintln!("[{}:{}] {} = {:?}",
                        file!(), line!(), stringify!($val), &tmp);
                    tmp
                }
            }
        };
        ($val:expr,) => { $crate::lg!($val) };
        ($($val:expr),+ $(,)?) => {
            ($($crate::lg!($val)),+,)
        };
    }

    #[macro_export]
    macro_rules! lg_nl {
        () => {
            eprintln!("[{}:{}]", $crate::file!(), $crate::line!());
        };
        ($val:expr) => {
            match $val {
                tmp => {
                    eprintln!("[{}:{}] {}:\n{:?}", file!(), line!(), stringify!($val), tmp);
                    tmp
                }
            };
        };
    }

    #[macro_export]
    macro_rules! msg {
        () => {
            compile_error!();
        };
        ($msg:expr) => {
            $crate::eprintln!("[{}:{}][{}]", $crate::file!(), $crate::line!(), $msg);
        };
        ($msg:expr, $val:expr) => {
            match $val {
                tmp => {
                    eprintln!("[{}:{}][{}] {} = {:?}",
                        file!(), line!(), $msg, stringify!($val), &tmp);
                    tmp
                }
            }
        };
        ($msg:expr, $val:expr,) => { msg!($msg, $val) };
        ($msg:expr, $($val:expr),+ $(,)?) => {
            ($(msg!($msg, $val)),+,)
        };
    }

    #[macro_export]
    macro_rules! tabular {
        ($val:expr) => {
            $crate::lg_nl!(crate::dbg::Tabular($val))
        };
    }

    #[macro_export]
    macro_rules! boolean_table {
        ($val:expr) => {
            $crate::lg_nl!(crate::dbg::BooleanTable($val));
        };
    }

    #[macro_export]
    macro_rules! boolean_slice {
        ($val:expr) => {
            $crate::lg!(crate::dbg::BooleanSlice($val));
        };
    }

    use std::fmt::{Debug, Formatter};

    #[derive(Clone)]
    pub struct Tabular<'a, T: Debug>(pub &'a [T]);
    impl<'a, T: Debug> Debug for Tabular<'a, T> {
        fn fmt(&self, f: &mut Formatter) -> std::fmt::Result {
            for i in 0..self.0.len() {
                writeln!(f, "{:2} | {:?}", i, &self.0[i])?;
            }
            Ok(())
        }
    }

    #[derive(Clone)]
    pub struct BooleanTable<'a>(pub &'a [Vec<bool>]);
    impl<'a> Debug for BooleanTable<'a> {
        fn fmt(&self, f: &mut Formatter) -> std::fmt::Result {
            for i in 0..self.0.len() {
                writeln!(f, "{:2} | {:?}", i, BooleanSlice(&self.0[i]))?;
            }
            Ok(())
        }
    }

    #[derive(Clone)]
    pub struct BooleanSlice<'a>(pub &'a [bool]);
    impl<'a> Debug for BooleanSlice<'a> {
        fn fmt(&self, f: &mut Formatter) -> std::fmt::Result {
            write!(
                f,
                "{}",
                self.0
                    .iter()
                    .map(|&b| if b { "1 " } else { "0 " })
                    .collect::<String>()
            )?;
            Ok(())
        }
    }
}
// }}}
// polynomial {{{
#[allow(dead_code)]
mod polynomial {
    #![warn(missing_docs)]

    use crate::type_traits::{One, Ring, Zero};

    mod arith {
        use super::Poly;
        use crate::type_traits::Ring;
        use std::ops::*;

        impl<T: Ring + Copy> Add for Poly<T> {
            type Output = Poly<T>;
            fn add(self, rhs: Self) -> Self::Output {
                let mut res = self.0.clone();
                if self.0.len() < rhs.0.len() {
                    res.resize(rhs.0.len(), T::zero());
                }
                res.iter_mut().zip(rhs.0.iter()).for_each(|(x, y)| {
                    *x += *y;
                });
                Self::remove_trailig_zeros(&mut res);
                Poly(res)
            }
        }

        impl<T: Ring + Copy> Sub for Poly<T>
        where
            T: Sub + SubAssign,
        {
            type Output = Poly<T>;
            fn sub(self, rhs: Self) -> Self::Output {
                let mut res = self.0.clone();
                if self.0.len() < rhs.0.len() {
                    res.resize(rhs.0.len(), T::zero());
                }
                res.iter_mut().zip(rhs.0.iter()).for_each(|(x, y)| {
                    *x -= *y;
                });
                Self::remove_trailig_zeros(&mut res);
                Poly(res)
            }
        }

        #[allow(clippy::suspicious_arithmetic_impl)]
        impl<T: Ring + Copy> Mul for Poly<T> {
            type Output = Poly<T>;
            fn mul(self, rhs: Self) -> Self::Output {
                if self.0.is_empty() && rhs.0.is_empty() {
                    Poly(Vec::new())
                } else {
                    let mut res = vec![T::zero(); self.0.len() + rhs.0.len() - 1];
                    for (i, &x) in self.0.iter().enumerate() {
                        for (j, &y) in rhs.0.iter().enumerate() {
                            res[i + j] += x * y;
                        }
                    }
                    Self::remove_trailig_zeros(&mut res);
                    Poly(res)
                }
            }
        }

        // TODO: このマクロを fp のものと共通化します。
        macro_rules! forward_assign_biop {
            ($(impl $trait:ident, $fn_assign:ident, $fn:ident)*) => {
                $(
                    impl<T: Ring + Copy> $trait for Poly<T> {
                        #[inline]
                        fn $fn_assign(&mut self, rhs: Self) {
                            *self = self.clone().$fn(&rhs);
                        }
                    }
                )*
            };
        }
        forward_assign_biop! {
            impl AddAssign, add_assign, add
            impl SubAssign, sub_assign, sub
            impl MulAssign, mul_assign, mul
        }

        // TODO: このマクロを fp のものと共通化します。
        macro_rules! forward_ref_binop {
            ($(impl $imp:ident, $method:ident)*) => {
                $(
                    impl<'a, T: Ring + Ring + Copy> $imp<Poly<T>> for &'a Poly<T> {
                        type Output = Poly<T>;
                        #[inline]
                        fn $method(self, other: Poly<T>) -> Self::Output {
                            $imp::$method(self.clone(), other.clone())
                        }
                    }

                    impl<'a, T: Ring + Copy> $imp<&'a Poly<T>> for Poly<T> {
                        type Output = Poly<T>;
                        #[inline]
                        fn $method(self, other: &Poly<T>) -> Self::Output {
                            $imp::$method(self.clone(), other.clone())
                        }
                    }

                    impl<'a, T: Ring + Copy> $imp<&'a Poly<T>> for &'a Poly<T> {
                        type Output = Poly<T>;
                        #[inline]
                        fn $method(self, other: &Poly<T>) -> Self::Output {
                            $imp::$method(self.clone(), other)
                        }
                    }
                )*
            };
        }
        forward_ref_binop! {
            impl Add, add
            impl Sub, sub
            impl Mul, mul
        }
    }

    #[derive(Debug, Clone, PartialEq, Eq)]
    pub struct Poly<T>(pub Vec<T>);

    impl<T: Ring + Copy> Poly<T> {
        pub fn new(mut src: Vec<T>) -> Self {
            Poly::remove_trailig_zeros(&mut src);
            Poly(src)
        }

        pub fn into_inner(self) -> Vec<T> {
            self.0
        }

        pub fn pow(mut self, mut b: u64) -> Poly<T> {
            let mut res = Poly::one();
            while b != 0 {
                if b % 2 == 1 {
                    res *= self.clone();
                }
                self *= self.clone();
                b /= 2
            }
            res
        }

        fn remove_trailig_zeros(a: &mut Vec<T>) {
            while a.last().map(T::is_zero).unwrap_or(false) {
                a.pop();
            }
        }
    }

    impl<T: Ring + Copy> Default for Poly<T> {
        fn default() -> Poly<T> {
            Poly(Vec::new())
        }
    }

    impl<T: Ring + Copy> Zero for Poly<T> {
        fn zero() -> Poly<T> {
            Poly(Vec::new())
        }
    }

    impl<T: Ring + Copy> One for Poly<T> {
        fn one() -> Poly<T> {
            Poly(vec![T::one()])
        }
    }
}
// }}}
// factorial {{{
#[allow(dead_code)]
mod factorial {
    use std::{fmt, iter, ops, slice};

    #[derive(Debug, Clone)]
    pub struct Factorial<T> {
        table: Vec<T>,
        inverse: Vec<T>,
    }

    impl<T> Factorial<T>
    where
        T: Clone + fmt::Debug + ops::Mul<Output = T> + ops::Div<Output = T>,
    {
        pub fn new(len: usize, from_usize: impl Fn(usize) -> T) -> Self {
            if len == 0 {
                Self {
                    table: Vec::new(),
                    inverse: Vec::new(),
                }
            } else {
                let mut table = vec![from_usize(1); len];
                for i in 1..len {
                    table[i] = table[i - 1].clone() * from_usize(i);
                }
                let mut inverse = vec![from_usize(1); len];
                inverse[len - 1] = from_usize(1) / table[len - 1].clone();
                for i in (1..len).rev() {
                    inverse[i - 1] = inverse[i].clone() * from_usize(i);
                }
                Self { table, inverse }
            }
        }

        pub fn inv(&self, i: usize) -> T {
            self.inverse[i].clone()
        }

        pub fn falling(&self, n: usize, k: usize) -> T {
            assert!(k <= n, "未対応です。0 がほしいでしょうか。");
            assert!(n < self.table.len(), "ちわーす。範囲外でーすｗｗｗ");
            self[n].clone() * self.inv(k)
        }

        pub fn binom(&self, n: usize, k: usize) -> T {
            assert!(k <= n, "未対応です。0 がほしいでしょうか。");
            assert!(n < self.table.len(), "ちわーす。範囲外でーすｗｗｗ");
            self[n].clone() * self.inv(k) * self.inv(n - k)
        }

        pub fn multichoose(&self, n: usize, k: usize) -> T {
            self.binom(n + k - 1, k - 1)
        }

        pub fn multinom(&self, a: &[usize]) -> T
        where
            T: iter::Product<T>,
        {
            let n = a.iter().sum::<usize>();
            assert!(n < self.table.len(), "ちわーす。範囲外でーすｗｗｗ");
            self[n].clone() * a.iter().map(|&x| self.inv(x)).product::<T>()
        }
    }

    impl<T, I> ops::Index<I> for Factorial<T>
    where
        I: slice::SliceIndex<[T]>,
    {
        type Output = I::Output;

        #[inline]
        fn index(&self, index: I) -> &I::Output {
            &self.table[index]
        }
    }
}
// }}}
// fp {{{
#[allow(dead_code)]
mod fp {
    #![warn(missing_docs)]
    use crate::type_traits::*;
    pub use aliases::*;
    use std::{cmp, fmt, iter, mem, ops::*};

    mod arith {
        use super::{Fp, Modable, Value};
        use crate::type_traits::*;
        use std::ops::*;

        impl<Mod: Modable> Add for Fp<Mod>
        where
            Mod::Output: Value,
        {
            type Output = Self;

            #[inline]
            fn add(self, rhs: Self) -> Self {
                Self(Self::normalize_from_the_bottom(
                    self.into_inner() + rhs.into_inner(),
                ))
            }
        }

        impl<Mod: Modable> Sub for Fp<Mod>
        where
            Mod::Output: Value,
        {
            type Output = Self;

            #[inline]
            fn sub(self, rhs: Self) -> Self {
                Self(Self::normalize_from_the_top(
                    self.into_inner() - rhs.into_inner(),
                ))
            }
        }

        impl<Mod: Modable> Mul for Fp<Mod>
        where
            Mod::Output: Value,
        {
            type Output = Self;

            #[inline]
            fn mul(self, rhs: Self) -> Self {
                Self::new(self.into_inner() * rhs.into_inner())
            }
        }

        #[allow(clippy::suspicious_arithmetic_impl)]
        impl<Mod: Modable> Div for Fp<Mod>
        where
            Mod::Output: Value,
        {
            type Output = Self;

            #[inline]
            fn div(self, rhs: Self) -> Self {
                self * rhs.inv()
            }
        }

        impl<Mod: Modable> Neg for Fp<Mod>
        where
            Mod::Output: Value,
        {
            type Output = Self;

            #[inline]
            fn neg(self) -> Self {
                if self.into_inner().is_zero() {
                    Self::zero()
                } else {
                    Self(Mod::VALUE - self.into_inner())
                }
            }
        }

        impl<Mod: Modable> Neg for &Fp<Mod>
        where
            Mod::Output: Value,
        {
            type Output = Fp<Mod>;

            #[inline]
            fn neg(self) -> Fp<Mod> {
                if self.into_inner().is_zero() {
                    Fp::zero()
                } else {
                    Fp(Mod::VALUE - self.into_inner())
                }
            }
        }

        macro_rules! forward_assign_biop {
            ($(impl $trait:ident, $fn_assign:ident, $fn:ident)*) => {
                $(
                    impl<Mod: Modable> $trait for Fp<Mod>
                    where
                        Mod::Output: Value
                    {
                        #[inline]
                        fn $fn_assign(&mut self, rhs: Self) {
                            *self = self.$fn(rhs);
                        }
                    }
                )*
            };
        }
        forward_assign_biop! {
            impl AddAssign, add_assign, add
            impl SubAssign, sub_assign, sub
            impl MulAssign, mul_assign, mul
            impl DivAssign, div_assign, div
        }

        macro_rules! forward_ref_binop {
            ($(impl $imp:ident, $method:ident)*) => {
                $(
                    impl<'a, Mod: Modable> $imp<Fp<Mod>> for &'a Fp<Mod>
                    where
                        Mod::Output:Value
                    {
                        type Output = Fp<Mod>;

                        #[inline]
                        fn $method(self, other: Fp<Mod>) -> Self::Output {
                            $imp::$method(*self, other)
                        }
                    }

                    impl<'a, Mod: Modable> $imp<&'a Fp<Mod>> for Fp<Mod>
                    where
                        Mod::Output:Value
                    {
                        type Output = Fp<Mod>;

                        #[inline]
                        fn $method(self, other: &Fp<Mod>) -> Self::Output {
                            $imp::$method(self, *other)
                        }
                    }

                    impl<'a, Mod: Modable> $imp<&'a Fp<Mod>> for &'a Fp<Mod>
                    where
                        Mod::Output:Value
                    {
                        type Output = Fp<Mod>;

                        #[inline]
                        fn $method(self, other: &Fp<Mod>) -> Self::Output {
                            $imp::$method(*self, *other)
                        }
                    }
                )*
            };
        }
        forward_ref_binop! {
            impl Add, add
            impl Sub, sub
            impl Mul, mul
            impl Div, div
        }
    }

    #[macro_export]
    macro_rules! fp_vec {
        () => (
            Vec::<$crate::Fp<_>>::new()
        );
        ($elem:expr; $n:expr) => (
            vec![$crate::Fp::new($elem); $n]
        );
        ($($x:expr),+ $(,)?) => (
            vec![$($crate::Fp::new($x),)+];
        );
    }

    #[derive(Debug, Clone, Copy, PartialEq, Eq)]
    pub struct Fp<Mod: Modable>(Mod::Output);

    impl<Mod: Modable> Fp<Mod>
    where
        Mod::Output: Value,
    {
        #[inline]
        pub fn new(src: Mod::Output) -> Self {
            Self(Self::normalize(src))
        }

        #[inline]
        pub fn frac(num: Mod::Output, den: Mod::Output) -> Self {
            Self::new(num) / Self::new(den)
        }

        #[inline]
        pub fn into_inner(self) -> Mod::Output {
            self.0
        }

        #[allow(clippy::many_single_char_names)]
        pub fn inv(self) -> Self {
            assert_ne!(
                self.into_inner(),
                Mod::Output::zero(),
                "さては 0 の逆元を取ろうとしていますね？"
            );
            let mut x = self.into_inner();
            let mut y = Mod::VALUE;
            let mut u = Mod::Output::one();
            let mut v = Mod::Output::zero();
            while x != Mod::Output::zero() {
                let q = y / x;
                y -= q * x;
                v -= q * u;
                mem::swap(&mut x, &mut y);
                mem::swap(&mut u, &mut v);
            }
            assert!(
                x == Mod::Output::zero()
                    && y == Mod::Output::one()
                    && (u == Mod::VALUE || u == -Mod::VALUE)
                    && (-Mod::VALUE < v && v < Mod::VALUE)
            );
            Self(Self::normalize_from_the_top(v))
        }

        pub fn pow(mut self, mut p: u64) -> Self {
            let mut ans = Self::one();
            while p != 0 {
                if p % 2 == 1 {
                    ans *= self;
                }
                self *= self;
                p /= 2;
            }
            ans
        }

        #[inline]
        fn normalize(src: Mod::Output) -> Mod::Output {
            Self::normalize_from_the_top(src % Mod::VALUE)
        }

        #[inline]
        fn normalize_from_the_bottom(src: Mod::Output) -> Mod::Output {
            if Mod::VALUE <= src {
                src - Mod::VALUE
            } else {
                src
            }
        }

        #[inline]
        fn normalize_from_the_top(src: Mod::Output) -> Mod::Output {
            if src < Mod::Output::zero() {
                src + Mod::VALUE
            } else {
                src
            }
        }
    }

    impl<Mod: Modable> Zero for Fp<Mod>
    where
        Mod::Output: Value,
    {
        #[inline]
        fn zero() -> Fp<Mod> {
            Fp::new(Mod::Output::zero())
        }
    }

    impl<Mod: Modable> One for Fp<Mod>
    where
        Mod::Output: Value,
    {
        #[inline]
        fn one() -> Fp<Mod> {
            Fp::new(Mod::Output::one())
        }
    }

    impl<Mod: Modable> iter::Sum<Fp<Mod>> for Fp<Mod>
    where
        Mod::Output: Value,
    {
        fn sum<I>(iter: I) -> Self
        where
            I: iter::Iterator<Item = Fp<Mod>>,
        {
            iter.fold(Fp::zero(), Add::add)
        }
    }

    impl<'a, Mod: 'a + Modable> iter::Sum<&'a Fp<Mod>> for Fp<Mod>
    where
        Mod::Output: Value,
    {
        fn sum<I>(iter: I) -> Self
        where
            I: iter::Iterator<Item = &'a Fp<Mod>>,
        {
            iter.fold(Fp::zero(), Add::add)
        }
    }

    impl<Mod: Modable> iter::Product<Fp<Mod>> for Fp<Mod>
    where
        Mod::Output: Value,
    {
        fn product<I>(iter: I) -> Self
        where
            I: iter::Iterator<Item = Fp<Mod>>,
        {
            iter.fold(Self::one(), Mul::mul)
        }
    }

    impl<'a, Mod: 'a + Modable> iter::Product<&'a Fp<Mod>> for Fp<Mod>
    where
        Mod::Output: Value,
    {
        fn product<I>(iter: I) -> Self
        where
            I: iter::Iterator<Item = &'a Fp<Mod>>,
        {
            iter.fold(Self::one(), Mul::mul)
        }
    }

    impl<Mod: Modable> fmt::Display for Fp<Mod>
    where
        Mod::Output: Value,
    {
        fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> Result<(), std::fmt::Error> {
            write!(f, "{}", self.0)
        }
    }

    pub trait Modable: Constant + Clone + fmt::Debug + cmp::PartialEq + cmp::Eq {}
    impl<Mod: Constant + Clone + fmt::Debug + cmp::PartialEq + cmp::Eq> Modable for Mod {}

    pub trait Value:
        Sized
        + Clone
        + Copy
        + Ring
        + fmt::Debug
        + fmt::Display
        + cmp::PartialOrd
        + cmp::Ord
        + cmp::Eq
        + Div<Output = Self>
        + Rem<Output = Self>
        + Neg<Output = Self>
        + DivAssign
        + RemAssign
    {
    }

    macro_rules! impl_value {
        ($($type:ty,)*) => { $(impl Value for $type {})* };
    }

    impl_value! { i8, i16, i32, i64, i128, isize, }

    mod aliases {
        use super::Fp;
        use crate::type_traits::Constant;

        #[derive(Debug, Clone, Copy, PartialEq, Eq)]
        pub struct Mod1000000007 {}
        pub type F1000000007 = Fp<Mod1000000007>;
        impl Constant for Mod1000000007 {
            type Output = i64;
            const VALUE: i64 = 1_000_000_007;
        }

        #[derive(Debug, Clone, Copy, PartialEq, Eq)]
        pub struct Mod998244353 {}
        pub type F998244353 = Fp<Mod998244353>;
        impl Constant for Mod998244353 {
            type Output = i64;
            const VALUE: i64 = 998_244_353;
        }
    }
}
// }}}
// type_traits {{{
#[allow(dead_code)]
mod type_traits {
    #![warn(missing_docs)]

    use std::{cmp, fmt, ops};

    mod primitive {
        use super::{
            binary::{Add, Mul},
            Commut, One, OpN, Zero,
        };

        macro_rules! zero {
            (int) => {
                0
            };
            (float) => {
                0.
            };
        }
        macro_rules! one {
            (int) => {
                1
            };
            (float) => {
                1.
            };
        }
        macro_rules! num {
            (@kind $kind:ident @type $T:ty) => {
                impl Zero for $T {
                    fn zero() -> $T {
                        zero!($kind)
                    }
                }
                impl One for $T {
                    fn one() -> $T {
                        one!($kind)
                    }
                }
                impl Commut for Add<$T> {}
                impl OpN for Add<$T> {
                    fn op_n(self, n: u64) -> Self {
                        Add(self.0 * n as $T)
                    }
                }
                impl Commut for Mul<$T> {}
            };
        }

        macro_rules! int {
            ($($T:ty,)*) => {
                $( num! { @kind int @type $T } )*
            };
        }

        macro_rules! float {
            ($($T:ty,)*) => {
                $( num! { @kind float @type $T } )*
            };
        }

        int! {
            u8, u16, u32, u64, u128, usize,
            i8, i16, i32, i64, i128, isize,
        }

        float! {
            f32, f64,
        }
    }

    pub mod binary {
        use super::{Assoc, Commut, Element, Identity, One, OpN, Peek, Zero};
        use std::ops;

        macro_rules! triv_wrapper {
            ($name:ident<$T:ident>) => {
                impl<$T> Peek for $name<$T>
                where
                    $T: Element,
                {
                    type Inner = $T;
                    fn peek(&self) -> $T {
                        self.0.clone()
                    }
                }
            };
        }

        #[derive(Debug, Clone, PartialEq, Eq, PartialOrd, Ord)]
        pub struct Len<T> {
            pub len: u64,
            pub base: T,
        }
        impl<T> Len<T> {
            pub fn single(base: T) -> Self {
                Len { len: 1, base }
            }
        }
        impl<T: Assoc> Assoc for Len<T> {
            fn op(self, rhs: Self) -> Self {
                Len {
                    len: self.len + rhs.len,
                    base: self.base.op(rhs.base),
                }
            }
        }

        #[derive(Debug, Clone, PartialEq, Eq, PartialOrd, Ord)]
        pub struct Pair<T, U>(pub T, pub U);
        impl<T: Assoc, U: Assoc> Assoc for Pair<T, U> {
            fn op(self, rhs: Self) -> Self {
                Pair(self.0.op(rhs.0), self.1.op(rhs.1))
            }
        }
        impl<T: Identity, U: Identity> Identity for Pair<T, U> {
            fn identity() -> Self {
                Pair(T::identity(), U::identity())
            }
        }
        impl<T: Commut, U: Commut> Commut for Pair<T, U> {}
        impl<T: OpN, U: OpN> OpN for Pair<T, U> {
            fn op_n(self, n: u64) -> Self {
                let Pair(t, u) = self;
                Self(t.op_n(n), u.op_n(n))
            }
        }

        #[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord)]
        pub struct BitXor<T>(pub T);
        triv_wrapper! { BitXor<T> }
        impl<T> Assoc for BitXor<T>
        where
            T: ops::BitXor<Output = T> + Element,
        {
            fn op(self, rhs: Self) -> Self {
                BitXor(self.0 ^ rhs.0)
            }
        }
        impl<T> Identity for BitXor<T>
        where
            T: ops::BitXor<Output = T> + Zero,
        {
            fn identity() -> Self {
                BitXor(T::zero())
            }
        }
        impl<T: ops::BitXor<Output = T> + Element> Commut for BitXor<T> {}
        impl<T: ops::BitXor<Output = T> + Zero> OpN for BitXor<T> {
            fn op_n(self, n: u64) -> Self {
                if n % 2 == 0 {
                    Self::identity()
                } else {
                    self
                }
            }
        }

        #[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord)]
        pub struct Add<T>(pub T);
        triv_wrapper! { Add<T> }
        impl<T> Assoc for Add<T>
        where
            T: ops::Add<Output = T> + Element,
        {
            fn op(self, rhs: Self) -> Self {
                Add(self.0 + rhs.0)
            }
        }
        impl<T> Identity for Add<T>
        where
            T: Zero,
        {
            fn identity() -> Self {
                Add(T::zero())
            }
        }

        #[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord)]
        pub struct Mul<T>(pub T);
        triv_wrapper! { Mul<T> }
        impl<T> Assoc for Mul<T>
        where
            T: ops::Mul<Output = T> + Element,
        {
            fn op(self, rhs: Self) -> Self {
                Mul(self.0 * rhs.0)
            }
        }
        impl<T> Identity for Mul<T>
        where
            T: One,
        {
            fn identity() -> Self {
                Mul(T::one())
            }
        }

        #[derive(Debug, Clone, Copy, PartialEq, Eq, Default)]
        pub struct InvertionNumber {
            pub zero: u64,
            pub one: u64,
            pub invertion: u64,
        }
        impl Peek for InvertionNumber {
            type Inner = bool;
            fn peek(&self) -> bool {
                self.try_get_bool()
                    .expect("ビットでないものを peek するのはいけません！")
            }
        }

        impl Assoc for InvertionNumber {
            fn op(self, rhs: Self) -> Self {
                InvertionNumber {
                    zero: self.zero + rhs.zero,
                    one: self.one + rhs.one,
                    invertion: self.invertion + self.one * rhs.zero + rhs.invertion,
                }
            }
        }
        impl Identity for InvertionNumber {
            fn identity() -> Self {
                InvertionNumber::new()
            }
        }
        impl InvertionNumber {
            pub fn new() -> Self {
                InvertionNumber {
                    zero: 0,
                    one: 0,
                    invertion: 0,
                }
            }
            pub fn try_get_u8(&self) -> Option<u8> {
                self.try_get_bool().map(|b| if b { 1 } else { 0 })
            }
            pub fn try_get_bool(&self) -> Option<bool> {
                if *self == Self::bit_zero() {
                    Some(false)
                } else if *self == Self::bit_one() {
                    Some(true)
                } else {
                    None
                }
            }
            pub fn count_zeros(&self) -> u64 {
                self.zero
            }
            pub fn count_ones(&self) -> u64 {
                self.one
            }
            pub fn inversion_number(&self) -> u64 {
                self.invertion
            }
            pub fn bit_zero() -> Self {
                InvertionNumber {
                    zero: 1,
                    one: 0,
                    invertion: 0,
                }
            }
            pub fn bit_one() -> Self {
                InvertionNumber {
                    zero: 0,
                    one: 1,
                    invertion: 0,
                }
            }
            pub fn from_bool(b: bool) -> Self {
                if b {
                    Self::bit_one()
                } else {
                    Self::bit_zero()
                }
            }
            pub fn from_u8(x: u8) -> Self {
                match x {
                    0 => Self::bit_zero(),
                    1 => Self::bit_one(),
                    _ => panic!("0 でも 1 でもないものを渡すとパニックです。"),
                }
            }
        }

        #[derive(Debug, Clone, Copy, PartialEq, Eq)]
        pub struct Affine<T> {
            pub a: T,
            pub b: T,
        }
        impl<T: Element> Peek for Affine<T> {
            type Inner = (T, T);
            fn peek(&self) -> (T, T) {
                (self.a.clone(), self.b.clone())
            }
        }
        impl<T> Assoc for Affine<T>
        where
            T: ops::Add<Output = T> + Element,
            T: ops::Mul<Output = T> + Element,
        {
            fn op(self, rhs: Self) -> Self {
                Self {
                    a: self.a.clone() * rhs.a,
                    b: self.b + self.a * rhs.b,
                }
            }
        }
        impl<T> Identity for Affine<T>
        where
            T: Zero + One,
        {
            fn identity() -> Self {
                Self {
                    a: T::one(),
                    b: T::zero(),
                }
            }
        }

        #[derive(Debug, Clone, PartialEq, Eq, PartialOrd, Ord)]
        pub struct Cat(pub String);
        impl Peek for Cat {
            type Inner = String;
            fn peek(&self) -> String {
                self.0.clone()
            }
        }
        impl Assoc for Cat {
            fn op(self, rhs: Self) -> Self {
                Cat(self.0.chars().chain(rhs.0.chars()).collect())
            }
        }
        impl Identity for Cat {
            fn identity() -> Self {
                Cat(String::new())
            }
        }
    }

    pub mod actions {
        use super::{binary::Len, Action, Commut, OpN};

        #[derive(Debug, Clone, PartialEq, Copy)]
        pub struct Update<T>(pub T);

        impl<T: OpN> Action for Update<T> {
            type Space = Len<T>;
            fn acted(self, x: Self::Space) -> Self::Space {
                Len {
                    len: x.len,
                    base: self.0.op_n(x.len),
                }
            }
        }

        #[derive(Debug, Clone, PartialEq, Copy)]
        pub struct Adjoint<T>(pub T);

        impl<T: OpN + Commut> Action for Adjoint<T> {
            type Space = Len<T>;
            fn acted(self, x: Self::Space) -> Self::Space {
                Len {
                    len: x.len,
                    base: self.0.op_n(x.len).op(x.base),
                }
            }
        }
    }

    pub trait Element: Sized + Clone + PartialEq + fmt::Debug {}
    impl<T: Sized + Clone + PartialEq + fmt::Debug> Element for T {}

    pub trait Assoc: Element {
        fn op(self, rhs: Self) -> Self;

        fn op_from_left(&mut self, left: &Self) {
            *self = Self::op(left.clone(), self.clone());
        }

        fn op_from_right(&mut self, right: &Self) {
            *self = Self::op(self.clone(), right.clone());
        }
    }

    pub trait Identity: Assoc {
        fn identity() -> Self;
    }

    pub trait Commut: Assoc {}

    pub trait OpN: Assoc {
        fn op_n(self, n: u64) -> Self;
    }

    #[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord)]
    pub struct Grade<T> {
        pub base: T,
        pub deg: usize,
    }
    impl<T: Assoc> Assoc for Grade<T> {
        fn op(self, rhs: Self) -> Self {
            Grade {
                deg: self.deg + rhs.deg,
                base: self.base.op(rhs.base),
            }
        }
    }

    pub trait Action {
        type Space: Assoc;
        fn acted(self, x: Self::Space) -> Self::Space;
    }

    pub trait Zero: ops::Add<Output = Self> + ops::AddAssign + Element {
        fn zero() -> Self;

        fn is_zero(&self) -> bool
        where
            Self: cmp::PartialEq,
        {
            self == &Self::zero()
        }
    }

    pub trait One: ops::Mul<Output = Self> + ops::MulAssign + Element {
        fn one() -> Self;

        fn is_one(&self) -> bool
        where
            Self: cmp::PartialEq,
        {
            self == &Self::one()
        }
    }

    pub trait Ring: Zero + One + ops::Neg + ops::Sub<Output = Self> + ops::SubAssign {}
    impl<T: Zero + One + ops::Neg + ops::Sub<Output = Self> + ops::SubAssign> Ring for T {}

    #[macro_export]
    macro_rules! define_constant {
        ($(#[$attr:meta])? $vis:vis type $wrapper_type:ident: $value_type:ty = $value:expr;) => {
            $(#[$attr])?
            #[derive(Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord)]
            $vis struct $wrapper_type {}

            impl Constant for $wrapper_type {
                type Output = $value_type;
                const VALUE: Self::Output = $value;
            }
        };
    }

    pub trait Peek {
        type Inner: fmt::Debug;

        fn peek(&self) -> Self::Inner;
    }

    pub trait Constant: Copy {
        type Output: Copy;

        const VALUE: Self::Output;
    }
}
// }}}
// fft {{{
#[allow(dead_code)]
mod fft {
    #![warn(missing_docs)]

    use crate::polynomial::Poly;
    use crate::type_traits::Ring;
    use std::{iter, marker};

    mod fftable {
        use super::Fftable;
        use crate::fp::F998244353;

        impl Fftable for F998244353 {
            fn root() -> F998244353 {
                F998244353::new(3).pow(7 * 17)
            }
            fn root_inv() -> F998244353 {
                F998244353::root().inv()
            }
            fn lg_ord() -> usize {
                23
            }
            fn div_assign_by_usize(&mut self, den: usize) {
                *self /= F998244353::new(den as i64)
            }
        }
    }

    pub trait Fftable: Ring + Copy {
        fn root() -> Self;

        fn root_inv() -> Self;

        fn lg_ord() -> usize;

        fn div_assign_by_usize(&mut self, den: usize);

        fn root_seq<Tag: DirectionTag>() -> Vec<Self> {
            let mut root = Tag::root::<Self>();
            let mut res = Vec::with_capacity(Self::lg_ord());
            for _ in 0..Self::lg_ord() {
                res.push(root);
                root *= root;
            }
            res.push(root);
            res.reverse();
            res
        }
    }

    #[must_use]
    pub fn multiply<T>(a: Poly<T>, b: Poly<T>) -> Poly<T>
    where
        T: Fftable + std::fmt::Debug,
    {
        let mut a = a.into_inner();
        let mut b = b.into_inner();
        let n = (a.len() + b.len()).next_power_of_two();
        a.resize(n, T::zero());
        b.resize(n, T::zero());
        a = fft(&a, marker::PhantomData::<Forward>);
        b = fft(&b, marker::PhantomData::<Forward>);
        let mut c = a
            .iter()
            .copied()
            .zip(b.iter().copied())
            .map(|(x, y)| x * y)
            .collect::<Vec<_>>();
        c = fft(&c, marker::PhantomData::<Backward>);
        c.iter_mut().for_each(|x| x.div_assign_by_usize(n));
        Poly::new(c)
    }

    #[allow(clippy::many_single_char_names)]
    #[must_use]
    pub fn fft<T, Tag>(a: &[T], _tag: marker::PhantomData<Tag>) -> Vec<T>
    where
        T: Fftable + std::fmt::Debug,
        Tag: DirectionTag,
    {
        let n = a.len();
        assert!(n.is_power_of_two());
        let mut a = bit_reverse(a);
        for (d, &root) in iter::successors(Some(1), |x| Some(x * 2))
            .take_while(|&d| d != n)
            .zip(T::root_seq::<Tag>()[1..].iter())
        {
            for (i, coeff) in iter::successors(Some((0, T::one())), |&(mut i, mut coeff)| {
                i += 1;
                coeff *= root;
                if i % d == 0 {
                    i += d;
                    coeff = T::one();
                }
                Some((i, coeff))
            })
            .take_while(|&(i, _)| i != n)
            {
                let x = a[i];
                let y = a[i + d];
                a[i] = x + y * coeff;
                a[i + d] = x - y * coeff;
            }
        }
        a
    }

    pub trait DirectionTag {
        fn root<T: Fftable>() -> T;
    }
    pub struct Forward {}
    impl DirectionTag for Forward {
        fn root<T: Fftable>() -> T {
            T::root()
        }
    }
    pub struct Backward {}
    impl DirectionTag for Backward {
        fn root<T: Fftable>() -> T {
            T::root_inv()
        }
    }

    #[must_use]
    fn bit_reverse<T: Copy>(a: &[T]) -> Vec<T> {
        assert!(a.len().is_power_of_two());
        let shift = std::mem::size_of::<usize>() as u32 * 8 - a.len().trailing_zeros();
        (0..a.len())
            .map(|i| a[if i == 0 { 0 } else { i.reverse_bits() >> shift }])
            .collect()
    }
}
// }}}
// ngtio {{{
#[allow(dead_code)]
mod ngtio {
    #![warn(missing_docs)]

    mod i {
        use std::{
            io::{self, BufRead},
            iter,
        };

        pub use multi_token::{Leaf, Parser, ParserTuple, RawTuple, Tuple, VecLen};
        pub use token::{Token, Usize1};

        pub fn with_stdin() -> Tokenizer<io::BufReader<io::Stdin>> {
            io::BufReader::new(io::stdin()).tokenizer()
        }

        pub fn with_str(src: &str) -> Tokenizer<&[u8]> {
            src.as_bytes().tokenizer()
        }

        pub struct Tokenizer<S: BufRead> {
            queue: Vec<String>, // FIXME: String のみにすると速そうです。
            scanner: S,
        }
        macro_rules! prim_method {
            ($name:ident: $T:ty) => {
#[allow(missing_docs)]
                pub fn $name(&mut self) -> $T {
                    <$T>::leaf().parse(self)
                }
            };
            ($name:ident) => {
                prim_method!($name: $name);
            };
        }
        macro_rules! prim_methods {
            ($name:ident: $T:ty; $($rest:tt)*) => {
                prim_method!($name:$T);
                prim_methods!($($rest)*);
            };
            ($name:ident; $($rest:tt)*) => {
                prim_method!($name);
                prim_methods!($($rest)*);
            };
            () => ()
        }
        impl<S: BufRead> Tokenizer<S> {
            pub fn token(&mut self) -> String {
                self.load();
                self.queue.pop().expect("入力が終了したのですが。")
            }
            pub fn new(scanner: S) -> Self {
                Self {
                    queue: Vec::new(),
                    scanner,
                }
            }
            fn load(&mut self) {
                while self.queue.is_empty() {
                    let mut s = String::new();
                    let length = self.scanner.read_line(&mut s).unwrap(); // 入力が UTF-8 でないときにエラーだそうです。
                    if length == 0 {
                        break;
                    }
                    self.queue = s.split_whitespace().rev().map(str::to_owned).collect();
                }
            }

            pub fn skip_line(&mut self) {
                assert!(
                    self.queue.is_empty(),
                    "行の途中で呼ばないでいただきたいです。現在のトークンキュー: {:?}",
                    &self.queue
                );
                self.load();
            }

            pub fn end(&mut self) {
                self.load();
                assert!(self.queue.is_empty(), "入力はまだあります！");
            }

            pub fn parse<T: Token>(&mut self) -> T::Output {
                T::parse(&self.token())
            }

            pub fn parse_collect<T: Token, B>(&mut self, n: usize) -> B
            where
                B: iter::FromIterator<T::Output>,
            {
                iter::repeat_with(|| self.parse::<T>()).take(n).collect()
            }

            pub fn tuple<T: RawTuple>(&mut self) -> <T::LeafTuple as Parser>::Output {
                T::leaf_tuple().parse(self)
            }

            pub fn vec<T: Token>(&mut self, len: usize) -> Vec<T::Output> {
                T::leaf().vec(len).parse(self)
            }

            pub fn vec_tuple<T: RawTuple>(
                &mut self,
                len: usize,
            ) -> Vec<<T::LeafTuple as Parser>::Output> {
                T::leaf_tuple().vec(len).parse(self)
            }

            pub fn vec2<T: Token>(&mut self, height: usize, width: usize) -> Vec<Vec<T::Output>> {
                T::leaf().vec(width).vec(height).parse(self)
            }

            pub fn vec2_tuple<T>(
                &mut self,
                height: usize,
                width: usize,
            ) -> Vec<Vec<<T::LeafTuple as Parser>::Output>>
            where
                T: RawTuple,
            {
                T::leaf_tuple().vec(width).vec(height).parse(self)
            }
            prim_methods! {
                u8; u16; u32; u64; u128; usize;
                i8; i16; i32; i64; i128; isize;
                char; string: String;
            }
        }

        mod token {
            use super::multi_token::Leaf;
            use std::{any, fmt, marker, str};

            pub trait Token: Sized {
                type Output;
                fn parse(s: &str) -> Self::Output;
                fn leaf() -> Leaf<Self> {
                    Leaf(marker::PhantomData)
                }
            }

            impl<T> Token for T
            where
                T: str::FromStr,
                <T as str::FromStr>::Err: fmt::Debug,
            {
                type Output = T;
                fn parse(s: &str) -> Self::Output {
                    s.parse().unwrap_or_else(|_| {
                        panic!("Parse error!: ({}: {})", s, any::type_name::<T>(),)
                    })
                }
            }

            pub struct Usize1 {}
            impl Token for Usize1 {
                type Output = usize;
                fn parse(s: &str) -> Self::Output {
                    usize::parse(s)
                        .checked_sub(1)
                        .expect("Parse error! (Zero substruction error of Usize1)")
                }
            }
        }

        mod multi_token {
            use super::{Token, Tokenizer};
            use std::{io::BufRead, iter, marker};

            pub trait Parser: Sized {
                type Output;
                fn parse<S: BufRead>(&self, server: &mut Tokenizer<S>) -> Self::Output;
                fn vec(self, len: usize) -> VecLen<Self> {
                    VecLen { len, elem: self }
                }
            }
            pub struct Leaf<T>(pub(super) marker::PhantomData<T>);
            impl<T: Token> Parser for Leaf<T> {
                type Output = T::Output;
                fn parse<S: BufRead>(&self, server: &mut Tokenizer<S>) -> T::Output {
                    server.parse::<T>()
                }
            }

            pub struct VecLen<T> {
                pub len: usize,
                pub elem: T,
            }
            impl<T: Parser> Parser for VecLen<T> {
                type Output = Vec<T::Output>;
                fn parse<S: BufRead>(&self, server: &mut Tokenizer<S>) -> Self::Output {
                    iter::repeat_with(|| self.elem.parse(server))
                        .take(self.len)
                        .collect()
                }
            }

            pub trait RawTuple {
                type LeafTuple: Parser;
                fn leaf_tuple() -> Self::LeafTuple;
            }
            pub trait ParserTuple {
                type Tuple: Parser;
                fn tuple(self) -> Self::Tuple;
            }
            pub struct Tuple<T>(pub T);
            macro_rules! impl_tuple {
                ($($t:ident: $T:ident),*) => {
                    impl<$($T),*> Parser for Tuple<($($T,)*)>
                        where
                            $($T: Parser,)*
                            {
                                type Output = ($($T::Output,)*);
#[allow(unused_variables)]
                                fn parse<S: BufRead >(&self, server: &mut Tokenizer<S>) -> Self::Output {
                                    match self {
                                        Tuple(($($t,)*)) => {
                                            ($($t.parse(server),)*)
                                        }
                                    }
                                }
                            }
                    impl<$($T: Token),*> RawTuple for ($($T,)*) {
                        type LeafTuple = Tuple<($(Leaf<$T>,)*)>;
                        fn leaf_tuple() -> Self::LeafTuple {
                            Tuple(($($T::leaf(),)*))
                        }
                    }
                    impl<$($T: Parser),*> ParserTuple for ($($T,)*) {
                        type Tuple = Tuple<($($T,)*)>;
                        fn tuple(self) -> Self::Tuple {
                            Tuple(self)
                        }
                    }
                };
            }
            impl_tuple!();
            impl_tuple!(t1: T1);
            impl_tuple!(t1: T1, t2: T2);
            impl_tuple!(t1: T1, t2: T2, t3: T3);
            impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4);
            impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5);
            impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6);
            impl_tuple!(t1: T1, t2: T2, t3: T3, t4: T4, t5: T5, t6: T6, t7: T7);
            impl_tuple!(
                t1: T1,
                t2: T2,
                t3: T3,
                t4: T4,
                t5: T5,
                t6: T6,
                t7: T7,
                t8: T8
            );
        }

        trait Scanner: BufRead + Sized {
            fn tokenizer(self) -> Tokenizer<Self> {
                Tokenizer::new(self)
            }
        }
        impl<R: BufRead> Scanner for R {}
    }

    pub use i::{with_stdin, with_str};

    pub mod prelude {
        pub use super::i::{Parser, ParserTuple, RawTuple, Token, Usize1};
    }
}
// }}}