type Mint = ModInt998244353;

fn main() {
    let mut sc = Scanner::new();
    let n: u64 = sc.next_num();
    let m: u64 = sc.next_num();

    let mut is_prime = vec![true; n as usize + 1];
    is_prime[1] = false;
    for i in 2..n + 1 {
        if !is_prime[i as usize] {
            continue;
        }
        let mut k = 2;
        while i * k <= n {
            is_prime[(i * k) as usize] = false;
            k += 1;
        }
    }

    let mut ans = Mint::from(0);
    let mut pow_m = Mint::from(1);
    let mut tmp = Mint::from(1);
    let mut l = 1;
    for i in 1..=n {
        if l <= m && is_prime[i as usize] {
            l *= i;
        }
        pow_m *= m;
        tmp *= m / l;
        ans += pow_m - tmp;
    }
    println!("{}", ans);
}

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

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

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

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

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

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

        /// # Parameters
        /// * `a` `0 <= a < m`
        /// * `b` `0 <= b < m`
        ///
        /// # Returns
        /// a * b % m
        #[allow(clippy::many_single_char_names)]
        pub(crate) fn mul(&self, a: u32, b: u32) -> u32 {
            mul_mod(a, b, self._m, self.im)
        }
    }

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

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        let mut z = a as u64;
        z *= b as u64;
        let x = (((z as u128) * (im as u128)) >> 64) as u64;
        let mut v = z.wrapping_sub(x.wrapping_mul(m as u64)) as u32;
        if m <= v {
            v = v.wrapping_add(m);
        }
        v
    }

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

    /// Reference:
    /// M. Forisek and J. Jancina,
    /// Fast Primality Testing for Integers That Fit into a Machine Word
    ///
    /// # Parameters
    /// * `n` `0 <= n`
    /* const */
    pub(crate) fn is_prime(n: i32) -> bool {
        let n = n as i64;
        match n {
            _ if n <= 1 => return false,
            2 | 7 | 61 => return true,
            _ if n % 2 == 0 => return false,
            _ => {}
        }
        let mut d = n - 1;
        while d % 2 == 0 {
            d /= 2;
        }
        for &a in &[2, 7, 61] {
            let mut t = d;
            let mut y = pow_mod(a, t, n as i32);
            while t != n - 1 && y != 1 && y != n - 1 {
                y = y * y % n;
                t <<= 1;
            }
            if y != n - 1 && t % 2 == 0 {
                return false;
            }
        }
        true
    }

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

    /// # Parameters
    /// * `b` `1 <= b`
    ///
    /// # Returns
    /// (g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
    /* const */
    #[allow(clippy::many_single_char_names)]
    pub(crate) fn inv_gcd(a: i64, b: i64) -> (i64, i64) {
        let a = safe_mod(a, b);
        if a == 0 {
            return (b, 0);
        }

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

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

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

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

    /// Compile time (currently not) primitive root
    /// @param m must be prime
    /// @return primitive root (and minimum in now)
    /* const */
    pub(crate) fn primitive_root(m: i32) -> i32 {
        match m {
            2 => return 1,
            167_772_161 => return 3,
            469_762_049 => return 3,
            754_974_721 => return 11,
            998_244_353 => return 3,
            _ => {}
        }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

                if p < 1_000_000 {
                    assert_eq!(
                        (0..p - 1)
                            .scan(1, |i, _| {
                                *i = *i * g % p;
                                Some(*i)
                            })
                            .collect::<HashSet<_>>()
                            .len() as i32,
                        p - 1
                    );
                }
            }
        }
    }
}
pub mod modint {
    //! Structs that treat the modular arithmetic.
    //!
    //! For most of the problems, It is sufficient to use [`ModInt1000000007`] or [`ModInt998244353`], which can be used as follows.
    //!
    //! ```
    //! use ac_library_rs::ModInt1000000007 as Mint; // rename to whatever you want
    //! use proconio::{input, source::once::OnceSource};
    //!
    //! input! {
    //!     from OnceSource::from("1000000006 2\n"),
    //!     a: Mint,
    //!     b: Mint,
    //! }
    //!
    //! println!("{}", a + b); // `1`
    //! ```
    //!
    //! If the modulus is not fixed, you can use [`ModInt`] as follows.
    //!
    //! ```
    //! use ac_library_rs::ModInt as Mint; // rename to whatever you want
    //! use proconio::{input, source::once::OnceSource};
    //!
    //! input! {
    //!     from OnceSource::from("3 3 7\n"),
    //!     a: u32,
    //!     b: u32,
    //!     m: u32,
    //! }
    //!
    //! Mint::set_modulus(m);
    //! let a = Mint::new(a);
    //! let b = Mint::new(b);
    //!
    //! println!("{}", a * b); // `2`
    //! ```
    //!
    //! # Major changes from the original ACL
    //!
    //! - Converted the struct names to PascalCase.
    //! - Renamed `mod` → `modulus`.
    //! - Moduli are `u32`, not `i32`.
    //! - Each `Id` does not have a identifier number. Instead, they explicitly own `&'static LocalKey<RefCell<Barrett>>`.
    //! - The type of the argument of `pow` is `u64`, not `i64`.
    //! - Modints implement `FromStr` and `Display`. Modints in the original ACL don't have `operator<<` or `operator>>`.
    //!
    //! [`ModInt1000000007`]: ./type.ModInt1000000007.html
    //! [`ModInt998244353`]: ./type.ModInt998244353.html
    //! [`ModInt`]: ./type.ModInt.html

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

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

    /// Represents _ℤ/mℤ_ where _m_ is a constant value.
    ///
    /// Corresponds to `atcoder::static_modint` in the original ACL.
    ///
    /// # Example
    ///
    /// ```
    /// use ac_library_rs::ModInt1000000007 as Mint;
    /// use proconio::{input, source::once::OnceSource};
    ///
    /// input! {
    ///     from OnceSource::from("1000000006 2\n"),
    ///     a: Mint,
    ///     b: Mint,
    /// }
    ///
    /// println!("{}", a + b); // `1`
    /// ```
    #[derive(Copy, Clone, Eq, PartialEq)]
    #[repr(transparent)]
    pub struct StaticModInt<M> {
        val: u32,
        phantom: PhantomData<fn() -> M>,
    }

    impl<M: Modulus> StaticModInt<M> {
        /// Returns the modulus, which is [`<M as Modulus>::VALUE`].
        ///
        /// Corresponds to `atcoder::static_modint::mod` in the original ACL.
        ///
        /// # Example
        ///
        /// ```
        /// use ac_library_rs::ModInt1000000007 as Mint;
        ///
        /// assert_eq!(1_000_000_007, Mint::modulus());
        /// ```
        ///
        /// [`<M as Modulus>::VALUE`]: ../trait.Modulus.html#associatedconstant.VALUE
        #[inline(always)]
        pub fn modulus() -> u32 {
            M::VALUE
        }

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

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

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

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

        /// Retruns the multiplicative inverse of `self`.
        ///
        /// Corresponds to `atcoder::static_modint::inv` in the original ACL.
        ///
        /// # Panics
        ///
        /// Panics if the multiplicative inverse does not exist.
        #[inline]
        pub fn inv(self) -> Self {
            if M::HINT_VALUE_IS_PRIME {
                if self.val() == 0 {
                    panic!("attempt to divide by zero");
                }
                debug_assert!(
                    internal_math::is_prime(M::VALUE.try_into().unwrap()),
                    "{} is not a prime number",
                    M::VALUE,
                );
                self.pow((M::VALUE - 2).into())
            } else {
                Self::inv_for_non_prime_modulus(self)
            }
        }
    }

    /// These methods are implemented for the struct.
    /// You don't need to `use` `ModIntBase` to call methods of `StaticModInt`.
    impl<M: Modulus> ModIntBase for StaticModInt<M> {
        #[inline(always)]
        fn modulus() -> u32 {
            Self::modulus()
        }

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

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

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

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

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

    /// Represents _1000000007_.
    #[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
    pub enum Mod1000000007 {}

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

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

    /// Represents _998244353_.
    #[derive(Copy, Clone, Ord, PartialOrd, Eq, PartialEq, Hash, Debug)]
    pub enum Mod998244353 {}

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

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

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

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

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

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

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

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

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

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

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

    /// These methods are implemented for the struct.
    /// You don't need to `use` `ModIntBase` to call methods of `DynamicModInt`.
    impl<I: Id> ModIntBase for DynamicModInt<I> {
        #[inline]
        fn modulus() -> u32 {
            Self::modulus()
        }

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

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

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

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

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

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

    /// Pair of _m_ and _ceil(2⁶⁴/m)_.
    pub struct Barrett {
        m: AtomicU32,
        im: AtomicU64,
    }

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

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

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

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

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

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

    /// A trait for [`StaticModInt`] and [`DynamicModInt`].
    ///
    /// Corresponds to `atcoder::internal::modint_base` in the original ACL.
    ///
    /// [`StaticModInt`]: ../struct.StaticModInt.html
    /// [`DynamicModInt`]: ../struct.DynamicModInt.html
    pub trait ModIntBase:
        Default
        + FromStr
        + From<i8>
        + From<i16>
        + From<i32>
        + From<i64>
        + From<i128>
        + From<isize>
        + From<u8>
        + From<u16>
        + From<u32>
        + From<u64>
        + From<u128>
        + From<usize>
        + Copy
        + Eq
        + Hash
        + fmt::Display
        + fmt::Debug
        + Neg<Output = Self>
        + Add<Output = Self>
        + Sub<Output = Self>
        + Mul<Output = Self>
        + Div<Output = Self>
        + AddAssign
        + SubAssign
        + MulAssign
        + DivAssign
    {
        /// Returns the modulus.
        ///
        /// Corresponds to `atcoder::static_modint::mod` and `atcoder::dynamic_modint::mod` in the original ACL.
        ///
        /// # Example
        ///
        /// ```
        /// use ac_library_rs::modint::ModIntBase;
        ///
        /// fn f<Z: ModIntBase>() {
        ///     let _: u32 = Z::modulus();
        /// }
        /// ```
        fn modulus() -> u32;

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

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

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

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

        /// Returns `self` to the power of `n`.
        ///
        /// Corresponds to `atcoder::static_modint::pow` and `atcoder::dynamic_modint::pow` in the original ACL.
        ///
        /// # Example
        ///
        /// ```
        /// use ac_library_rs::modint::ModIntBase;
        ///
        /// fn f<Z: ModIntBase>() {
        ///     let _: Z = Z::new(2).pow(3);
        /// }
        /// ```
        #[inline]
        fn pow(self, mut n: u64) -> Self {
            let mut x = self;
            let mut r = Self::raw(1);
            while n > 0 {
                if n & 1 == 1 {
                    r *= x;
                }
                x *= x;
                n >>= 1;
            }
            r
        }
    }

    /// A trait for `{StaticModInt, DynamicModInt, ModIntBase}::new`.
    pub trait RemEuclidU32 {
        /// Calculates `self` _mod_ `modulus` losslessly.
        fn rem_euclid_u32(self, modulus: u32) -> u32;
    }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    impl_folding! {
        impl<M: Modulus> Sum<_>     for StaticModInt<M>  { fn sum(_)     -> _ { _(Self::raw(0), Add::add) } }
        impl<M: Modulus> Product<_> for StaticModInt<M>  { fn product(_) -> _ { _(Self::raw(1), Mul::mul) } }
        impl<I: Id     > Sum<_>     for DynamicModInt<I> { fn sum(_)     -> _ { _(Self::raw(0), Add::add) } }
        impl<I: Id     > Product<_> for DynamicModInt<I> { fn product(_) -> _ { _(Self::raw(1), Mul::mul) } }
    }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        #[test]
        fn static_modint_assign_coercion() {
            let f = ModInt1000000007::new;
            let a = f(10_293_812_usize);
            let b = 9_083_240_982_usize;
            let expected = (((a + b) * b) - b) / b;
            let mut c = a;
            c += b;
            c *= b;
            c -= b;
            c /= b;
            assert_eq!(expected, c);
        }
    }
}
use modint::*;
pub mod scanner {
    use std::{
        fmt::Debug,
        io::{stdin, Read},
        str::{FromStr, SplitWhitespace},
    };

    pub struct Scanner<'a> {
        ptr: *mut String,
        iter: SplitWhitespace<'a>,
    }

    impl Scanner<'_> {
        pub fn new() -> Scanner<'static> {
            let mut buf = String::new();
            stdin().read_to_string(&mut buf).unwrap();
            let ptr = Box::leak(Box::new(buf));
            Scanner {
                ptr,
                iter: ptr.split_whitespace(),
            }
        }

        pub fn next_token<T>(&mut self) -> T
        where
            T: Ord + FromStr,
            <T as FromStr>::Err: Debug,
        {
            self.iter.next().unwrap().parse().unwrap()
        }

        pub fn next_vec<T>(&mut self, size: usize) -> Vec<T>
        where
            T: Ord + FromStr,
            <T as FromStr>::Err: Debug,
        {
            let mut res = Vec::with_capacity(size);
            (0..size).for_each(|_| res.push(self.next_token()));
            res
        }
        pub fn next_chars(&mut self) -> Vec<char> {
            self.iter.next().unwrap().chars().collect()
        }
        pub fn next_bytes(&mut self) -> Vec<u8> {
            self.iter.next().unwrap().bytes().collect()
        }

        pub fn next_num<T>(&mut self) -> T
        where
            T: Ord + FromStr,
            <T as FromStr>::Err: Debug,
        {
            self.iter.next().unwrap().parse().unwrap()
        }
    }

    impl Drop for Scanner<'_> {
        fn drop(&mut self) {
            let _x = unsafe { Box::from_raw(self.ptr) };
        }
    }
}

pub mod util {
    pub trait SetMinMax {
        fn set_min(&mut self, v: Self) -> bool;
        fn set_max(&mut self, v: Self) -> bool;
    }
    impl<T: PartialOrd> SetMinMax for T {
        fn set_min(&mut self, v: T) -> bool {
            *self > v && {
                *self = v;
                true
            }
        }
        fn set_max(&mut self, v: T) -> bool {
            *self < v && {
                *self = v;
                true
            }
        }
    }

    #[macro_export]
    macro_rules! div_floor {
        ($a:expr, $b:expr) => {
            if $b < 0 {
                if $a <= 0 {
                    (-$a) / (-$b)
                } else {
                    (-$a + 1) / (-$b) - 1
                }
            } else {
                if $a >= 0 {
                    $a / $b
                } else {
                    ($a + 1) / $b - 1
                }
            }
        };
    }
    #[macro_export]
    macro_rules! div_ceil {
        ($a:expr, $b:expr) => {
            if $b < 0 {
                if $a < 0 {
                    (-$a - 1) / (-$b) + 1
                } else {
                    (-$a) / (-$b)
                }
            } else {
                if $a > 0 {
                    ($a - 1) / $b + 1
                } else {
                    $a / $b
                }
            }
        };
    }
}

use scanner::*;
use util::*;


fn main() {
    input! {
        n: usize,
        m: i64,
    }
    let p = sieve(n);
    let mut dp = vec![Z::new(0); n + 1];
    dp[0] = Z::new(1);
    let mut f = m;
    let mut ans = Z::new(0);
    for i in 1..=n {
        if p[i] {
            f /= i as i64;
        }
        dp[i] = Z::new(f) * dp[i - 1];
        ans += Z::new(m).pow(i as u32) - dp[i];
    }
    println!("{}", ans);
}

pub fn sieve(n: usize) -> Vec<bool> {
    let mut p = vec![true; n + 1];
    p[0] = false;
    p[1] = false;
    for i in 2..=n {
        if p[i] && i * i <= n {
            for j in (i * i..=n).step_by(i) {
                p[j] = false;
            }
        }
    }
    p
}

#[derive(Copy, Clone, Default)]
pub struct Z(i64);

impl Z {
    const MOD: i64 = 998_244_353;

    pub fn new(mut x: i64) -> Z {
        x %= Z::MOD;
        if x < 0 {
            x += Z::MOD;
        }
        Z(x)
    }

    pub fn pow(self, mut exp: u32) -> Z {
        let mut ans = Z::new(1);
        let mut base = self;
        while exp > 1 {
            if exp % 2 == 1 {
                ans = ans * base;
            }
            base = base * base;
            exp >>= 1;
        }
        ans * base
    }

    pub fn inv(self) -> Z {
        assert_ne!(self.0, 0);
        self.pow((Z::MOD - 2) as u32)
    }
}

impl std::ops::Neg for Z {
    type Output = Z;

    fn neg(self) -> Z {
        Z::new(-self.0)
    }
}

impl std::ops::Add<Z> for Z {
    type Output = Z;

    fn add(self, rhs: Z) -> Z {
        Z::new(self.0 + rhs.0)
    }
}

impl std::ops::Sub<Z> for Z {
    type Output = Z;

    fn sub(self, rhs: Z) -> Z {
        Z::new(self.0 - rhs.0)
    }
}

impl std::ops::Mul<Z> for Z {
    type Output = Z;

    fn mul(self, rhs: Z) -> Z {
        Z::new(self.0 * rhs.0)
    }
}

impl std::ops::Div<Z> for Z {
    type Output = Z;

    fn div(self, rhs: Z) -> Z {
        self * rhs.inv()
    }
}

impl std::ops::AddAssign<Z> for Z {
    fn add_assign(&mut self, rhs: Z) {
        *self = *self + rhs;
    }
}

impl std::ops::SubAssign<Z> for Z {
    fn sub_assign(&mut self, rhs: Z) {
        *self = *self - rhs;
    }
}

impl std::ops::MulAssign<Z> for Z {
    fn mul_assign(&mut self, rhs: Z) {
        *self = *self * rhs;
    }
}

impl std::ops::DivAssign<Z> for Z {
    fn div_assign(&mut self, rhs: Z) {
        *self = *self / rhs;
    }
}

impl std::fmt::Display for Z {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        self.0.fmt(f)
    }
}

impl std::fmt::Debug for Z {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        self.0.fmt(f)
    }
}

impl std::str::FromStr for Z {
    type Err = std::num::ParseIntError;

    fn from_str(s: &str) -> Result<Self, Self::Err> {
        Ok(Z::new(s.parse()?))
    }
}

mod io {
    use std::cell::RefCell;
    use std::io::*;

    std::thread_local! {
        pub static STDIN: RefCell<Stdin> = RefCell::new(stdin());
        pub static STDOUT: RefCell<BufWriter<Stdout>> = RefCell::new(BufWriter::new(stdout()));
    }

    #[macro_export]
    macro_rules! input {
        () => {};
        (mut $var:ident: $t:tt, $($rest:tt)*) => {
            let mut $var = __input_inner!($t);
            input!($($rest)*)
        };
        ($var:ident: $t:tt, $($rest:tt)*) => {
            let $var = __input_inner!($t);
            input!($($rest)*)
        };
        (mut $var:ident: $t:tt) => {
            let mut $var = __input_inner!($t);
        };
        ($var:ident: $t:tt) => {
            let $var = __input_inner!($t);
        };
    }

    #[macro_export]
    macro_rules! __input_inner {
        (($($t:tt),*)) => {
            ($(__input_inner!($t)),*)
        };
        ([$t:tt; $n:expr]) => {
            (0..$n).map(|_| __input_inner!($t)).collect::<Vec<_>>()
        };
        ([$t:tt]) => {{
            let n = __input_inner!(usize);
            (0..n).map(|_| __input_inner!($t)).collect::<Vec<_>>()
        }};
        (chars) => {
            __input_inner!(String).chars().collect::<Vec<_>>()
        };
        (bytes) => {
            __input_inner!(String).into_bytes()
        };
        (usize1) => {
            __input_inner!(usize) - 1
        };
        ($t:ty) => {
            $crate::io::STDIN.with(|r| {
                use std::io::BufRead;
                let r = r.borrow_mut();
                let mut r = r.lock();
                let mut s = vec![];
                loop {
                    let buf = r.fill_buf().unwrap();
                    if buf.is_empty() {
                        break;
                    }
                    if let Some(i) = buf.iter().position(u8::is_ascii_whitespace) {
                        s.extend_from_slice(&buf[..i]);
                        r.consume(i + 1);
                        if !s.is_empty() {
                            break;
                        }
                    } else {
                        s.extend_from_slice(buf);
                        let n = buf.len();
                        r.consume(n);
                    }
                }
                std::str::from_utf8(&s).unwrap().parse::<$t>().unwrap()
            })
        };
    }

    #[macro_export]
    macro_rules! println {
        () => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                std::writeln!(w.borrow_mut()).unwrap()
            })
        };
        ($($arg:tt)*) => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                std::writeln!(w.borrow_mut(), $($arg)*).unwrap()
            })
        };
    }

    #[macro_export]
    macro_rules! print {
        ($($arg:tt)*) => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                std::write!(w.borrow_mut(), $($arg)*).unwrap()
            })
        };
    }

    #[macro_export]
    macro_rules! flush {
        () => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                w.borrow_mut().flush().unwrap()
            })
        };
    }
}


const MOD: i64 = 998244353;

fn main() {
    input! {
        n: usize,
        m: i64,
    }
    let p = sieve(n);
    let mut ans = 0;
    let mut dp = vec![0; n + 1];
    dp[0] = 1;
    let mut f = 1;
    for i in 1..=n {
        if p[i] {
            f *= i as i64;
            f = f.min(m + 1);
        }
        dp[i] = m / f;
        dp[i] %= MOD;
        dp[i] *= dp[i - 1];
        dp[i] %= MOD;
        ans += modpow(m, i as i64);
        ans %= MOD;
        ans += MOD - dp[i];
        ans %= MOD;
    }
    println!("{}", ans);
}

pub fn sieve(n: usize) -> Vec<bool> {
    let mut p = vec![true; n + 1];
    p[0] = false;
    p[1] = false;
    for i in 2..=n {
        if p[i] && i * i <= n {
            for j in (i * i..=n).step_by(i) {
                p[j] = false;
            }
        }
    }
    p
}

pub fn modpow(mut base: i64, mut exp: i64) -> i64 {
    let mut ans = 1;
    base %= MOD;
    while exp > 0 {
        if exp % 2 == 1 {
            ans = ans * base % MOD;
        }
        base = base * base % MOD;
        exp >>= 1;
    }
    ans
}

mod io {
    use std::cell::RefCell;
    use std::io::*;

    std::thread_local! {
        pub static STDIN: RefCell<Stdin> = RefCell::new(stdin());
        pub static STDOUT: RefCell<BufWriter<Stdout>> = RefCell::new(BufWriter::new(stdout()));
    }

    #[macro_export]
    macro_rules! input {
        () => {};
        (mut $var:ident: $t:tt, $($rest:tt)*) => {
            let mut $var = __input_inner!($t);
            input!($($rest)*)
        };
        ($var:ident: $t:tt, $($rest:tt)*) => {
            let $var = __input_inner!($t);
            input!($($rest)*)
        };
        (mut $var:ident: $t:tt) => {
            let mut $var = __input_inner!($t);
        };
        ($var:ident: $t:tt) => {
            let $var = __input_inner!($t);
        };
    }

    #[macro_export]
    macro_rules! __input_inner {
        (($($t:tt),*)) => {
            ($(__input_inner!($t)),*)
        };
        ([$t:tt; $n:expr]) => {
            (0..$n).map(|_| __input_inner!($t)).collect::<Vec<_>>()
        };
        ([$t:tt]) => {{
            let n = __input_inner!(usize);
            (0..n).map(|_| __input_inner!($t)).collect::<Vec<_>>()
        }};
        (chars) => {
            __input_inner!(String).chars().collect::<Vec<_>>()
        };
        (bytes) => {
            __input_inner!(String).into_bytes()
        };
        (usize1) => {
            __input_inner!(usize) - 1
        };
        ($t:ty) => {
            $crate::io::STDIN.with(|r| {
                use std::io::BufRead;
                let r = r.borrow_mut();
                let mut r = r.lock();
                let mut s = vec![];
                loop {
                    let buf = r.fill_buf().unwrap();
                    if buf.is_empty() {
                        break;
                    }
                    if let Some(i) = buf.iter().position(u8::is_ascii_whitespace) {
                        s.extend_from_slice(&buf[..i]);
                        r.consume(i + 1);
                        if !s.is_empty() {
                            break;
                        }
                    } else {
                        s.extend_from_slice(buf);
                        let n = buf.len();
                        r.consume(n);
                    }
                }
                std::str::from_utf8(&s).unwrap().parse::<$t>().unwrap()
            })
        };
    }

    #[macro_export]
    macro_rules! println {
        () => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                std::writeln!(w.borrow_mut()).unwrap()
            })
        };
        ($($arg:tt)*) => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                std::writeln!(w.borrow_mut(), $($arg)*).unwrap()
            })
        };
    }

    #[macro_export]
    macro_rules! print {
        ($($arg:tt)*) => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                std::write!(w.borrow_mut(), $($arg)*).unwrap()
            })
        };
    }

    #[macro_export]
    macro_rules! flush {
        () => {
            $crate::io::STDOUT.with(|w| {
                use std::io::Write;
                w.borrow_mut().flush().unwrap()
            })
        };
    }
}


#![allow(unused_imports)]
use input::*;
use std::{
    collections::*,
    io::{self, BufWriter, Write},
};
fn run<I: Input, O: Write>(mut ss: I, mut out: O) {
    let t: u32 = 1;
    for _ in 0..t {
        case(&mut ss, &mut out);
    }
}
def_mint!(998244353);
fn case<I: Input, O: Write>(mut ss: I, mut out: O) {
    use mod_int::*;
    let (n, m): (usize, i64) = ss.parse();
    let ps = primes::primes(n);
    let mut ans = mint(0);
    let mut c = mint(1);
    let mut x = 1i64;
    let mut j = 0;
    for i in 1..=n {
        if j < ps.len() && i == ps[j] {
            x = x.saturating_mul(ps[j] as i64);
            j += 1;
        }
        c *= Mint::from((m / x) as u64);
        ans += Mint::from(m as u64).pow(i) - c;
    }
    wln!(out, "{}", ans);
}
fn main() {
    let stdin = io::stdin();
    let ss = SplitWs::new(stdin.lock());
    let stdout = io::stdout();
    let out = BufWriter::new(stdout.lock());
    run(ss, out);
}
pub mod mod_int {
    use std::{
        cmp,
        fmt::{self, Debug, Display},
        hash::Hash,
        iter::{Product, Sum},
        marker::PhantomData,
        mem,
        ops::*,
    };
    pub struct ModInt<M> {
        x: u32,
        marker: PhantomData<*const M>,
    }
    pub trait Modulo {
        fn modulo() -> u32;
    }
    impl<M> ModInt<M> {
        pub fn new(x: u32) -> Self {
            Self {
                x,
                marker: PhantomData,
            }
        }
        pub fn get(self) -> u32 {
            self.x
        }
    }
    impl<M: Modulo> ModInt<M> {
        pub fn modulo() -> u32 {
            M::modulo()
        }
        pub fn normalize(self) -> Self {
            Self::new(self.x % M::modulo())
        }
        pub fn inv(self) -> Self {
            assert_ne!(self.get(), 0);
            self.pow(M::modulo() - 2)
        }
        pub fn twice(self) -> Self {
            self + self
        }
        pub fn half(self) -> Self {
            if self.x & 1 == 0 {
                Self::new(self.x >> 1)
            } else {
                Self::new((self.x >> 1) + ((Self::modulo() + 1) >> 1))
            }
        }
    }
    impl<M> Clone for ModInt<M> {
        fn clone(&self) -> Self {
            Self::new(self.x)
        }
    }
    impl<M> Copy for ModInt<M> {}
    impl<M: Modulo> Neg for ModInt<M> {
        type Output = Self;
        fn neg(self) -> Self {
            Self::new(if self.x != 0 { M::modulo() - self.x } else { 0 })
        }
    }
    impl<M: Modulo> Neg for &ModInt<M> {
        type Output = ModInt<M>;
        fn neg(self) -> Self::Output {
            -(*self)
        }
    }
    impl<M: Modulo> Add for ModInt<M> {
        type Output = Self;
        fn add(self, rhs: Self) -> Self {
            let x = self.x + rhs.x;
            Self::new(if x < M::modulo() { x } else { x - M::modulo() })
        }
    }
    impl<M: Modulo> Sub for ModInt<M> {
        type Output = Self;
        fn sub(self, rhs: Self) -> Self {
            let x = if self.x >= rhs.x {
                self.x - rhs.x
            } else {
                M::modulo() + self.x - rhs.x
            };
            Self::new(x)
        }
    }
    impl<M: Modulo> Mul for ModInt<M> {
        type Output = Self;
        fn mul(self, rhs: Self) -> Self {
            let x = (self.x as u64 * rhs.x as u64) % M::modulo() as u64;
            Self::new(x as u32)
        }
    }
    impl<M: Modulo> Div for ModInt<M> {
        type Output = Self;
        #[allow(clippy::suspicious_arithmetic_impl)]
        fn div(self, rhs: Self) -> Self {
            self * rhs.inv()
        }
    }
    macro_rules! biops {
        ($ Op : ident , $ op : ident , $ OpAssign : ident , $ op_assign : ident) => {
            impl<M: Modulo> $Op<&Self> for ModInt<M> {
                type Output = Self;
                fn $op(self, rhs: &Self) -> Self {
                    self.$op(*rhs)
                }
            }
            impl<M: Modulo> $Op<ModInt<M>> for &ModInt<M> {
                type Output = ModInt<M>;
                fn $op(self, rhs: ModInt<M>) -> ModInt<M> {
                    (*self).$op(rhs)
                }
            }
            impl<M: Modulo> $Op for &ModInt<M> {
                type Output = ModInt<M>;
                fn $op(self, rhs: Self) -> ModInt<M> {
                    (*self).$op(*rhs)
                }
            }
            impl<M: Modulo> $OpAssign for ModInt<M> {
                fn $op_assign(&mut self, rhs: Self) {
                    *self = self.$op(rhs);
                }
            }
            impl<M: Modulo> $OpAssign<&Self> for ModInt<M> {
                fn $op_assign(&mut self, rhs: &Self) {
                    *self = self.$op(rhs);
                }
            }
        };
    }
    biops!(Add, add, AddAssign, add_assign);
    biops!(Sub, sub, SubAssign, sub_assign);
    biops!(Mul, mul, MulAssign, mul_assign);
    biops!(Div, div, DivAssign, div_assign);
    impl<M: Modulo> Sum for ModInt<M> {
        fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(ModInt::new(0), |x, y| x + y)
        }
    }
    impl<M: Modulo> Product for ModInt<M> {
        fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(ModInt::new(1), |x, y| x * y)
        }
    }
    macro_rules! fold {
        ($ Trait : ident , $ f : ident) => {
            impl<'a, M: Modulo + 'a> $Trait<&'a ModInt<M>> for ModInt<M> {
                fn $f<I: Iterator<Item = &'a ModInt<M>>>(iter: I) -> Self {
                    iter.copied().$f()
                }
            }
        };
    }
    fold!(Sum, sum);
    fold!(Product, product);
    pub trait Pow<Exp> {
        fn pow(self, exp: Exp) -> Self;
    }
    macro_rules! pow {
        ($ uty : ident , $ ity : ident) => {
            impl<M: Modulo> Pow<$uty> for ModInt<M> {
                fn pow(self, mut exp: $uty) -> Self {
                    if exp == 0 {
                        return ModInt::new(1);
                    }
                    let mut res = ModInt::new(1);
                    let mut base = self;
                    while exp > 1 {
                        if exp & 1 != 0 {
                            res *= base;
                        }
                        exp >>= 1;
                        base *= base;
                    }
                    base * res
                }
            }
            impl<M: Modulo> Pow<$ity> for ModInt<M> {
                fn pow(self, exp: $ity) -> Self {
                    if exp >= 0 {
                        self.pow(exp as $uty)
                    } else {
                        self.inv().pow(-exp as $uty)
                    }
                }
            }
        };
    }
    macro_rules ! impls { ($ m : ident , $ ($ uty : ident , $ ity : ident) ,*) => { $ ($ m ! ($ uty , $ ity) ;) * } ; }
    impls!(pow, usize, isize, u8, i8, u16, i16, u32, i32, u64, i64, u128, i128);
    impl<M> Default for ModInt<M> {
        fn default() -> Self {
            Self::new(0)
        }
    }
    impl<M> PartialEq for ModInt<M> {
        fn eq(&self, other: &Self) -> bool {
            self.x == other.x
        }
    }
    impl<M> Eq for ModInt<M> {}
    impl<M> PartialOrd for ModInt<M> {
        fn partial_cmp(&self, other: &Self) -> Option<cmp::Ordering> {
            self.x.partial_cmp(&other.x)
        }
    }
    impl<M> Ord for ModInt<M> {
        fn cmp(&self, other: &Self) -> cmp::Ordering {
            self.x.cmp(&other.x)
        }
    }
    impl<M> Hash for ModInt<M> {
        fn hash<H: std::hash::Hasher>(&self, state: &mut H) {
            self.x.hash(state)
        }
    }
    macro_rules! from_uint {
        ($ ty : ident) => {
            impl<M: Modulo> From<$ty> for ModInt<M> {
                fn from(x: $ty) -> Self {
                    if mem::size_of::<$ty>() <= 4 {
                        if ($ty::max_value() as u32) < M::modulo() {
                            Self::new(x as u32)
                        } else {
                            Self::new(x as u32).normalize()
                        }
                    } else {
                        Self::new((x % M::modulo() as $ty) as u32)
                    }
                }
            }
        };
    }
    macro_rules ! impls { ($ m : ident , $ ($ ty : ident) ,*) => { $ ($ m ! ($ ty) ;) * } ; }
    impls!(from_uint, usize, u8, u16, u32, u64, u128);
    macro_rules! from_small_int {
        ($ ty : ident) => {
            impl<M: Modulo> From<$ty> for ModInt<M> {
                fn from(x: $ty) -> Self {
                    let mut x = x as i32;
                    if x >= 0 {
                        Self::from(x as u32)
                    } else {
                        while x < 0 {
                            x += M::modulo() as i32;
                        }
                        Self::new(x as u32)
                    }
                }
            }
        };
    }
    impls!(from_small_int, i8, i16, i32);
    impl<M> Display for ModInt<M> {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            Display::fmt(&self.x, f)
        }
    }
    impl<M> Debug for ModInt<M> {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            Debug::fmt(&self.x, f)
        }
    }
    #[macro_export]
    macro_rules! def_mint {
        ($ modulo : expr) => {
            #[derive(Clone, Copy, PartialEq, Eq, Default, Debug)]
            pub struct MintModulo;
            impl crate::mod_int::Modulo for MintModulo {
                fn modulo() -> u32 {
                    $modulo
                }
            }
            pub type Mint = crate::mod_int::ModInt<MintModulo>;
            pub fn mint(x: u32) -> Mint {
                x.into()
            }
        };
    }
}
pub mod primes {
    pub fn primes(n: usize) -> Vec<usize> {
        const SKIP: [u8; 8] = [6, 4, 2, 4, 2, 4, 6, 2];
        const XTOI: [u8; 15] = [0, 0, 0, 1, 0, 2, 3, 0, 4, 5, 0, 6, 0, 0, 7];
        let mut sieve = vec![0u8; n / 30 + 1];
        let mut ps = vec![2, 3, 5];
        if n <= 4 {
            ps.truncate([0, 0, 1, 2, 2][n]);
            return ps;
        }
        let mut x = 7;
        let mut i = 1;
        while x <= n {
            if sieve[i / 8] & 1 << i % 8 == 0 {
                ps.push(x);
                let mut j = i;
                let mut y = x * x;
                while y <= n {
                    sieve[y / 30] |= 1 << XTOI[y / 2 % 15];
                    y += x * SKIP[j % 8] as usize;
                    j += 1;
                }
            }
            x += SKIP[i % 8] as usize;
            i += 1;
        }
        ps
    }
}
pub mod input {
    use std::{
        io::{self, prelude::*},
        marker::PhantomData,
    };
    pub trait Input {
        fn bytes(&mut self) -> &[u8];
        fn bytes_vec(&mut self) -> Vec<u8> {
            self.bytes().to_vec()
        }
        fn str(&mut self) -> &str {
            std::str::from_utf8(self.bytes()).unwrap()
        }
        fn parse<T: Parse>(&mut self) -> T {
            self.parse_with(DefaultParser)
        }
        fn parse_with<T>(&mut self, mut parser: impl Parser<T>) -> T {
            parser.parse(self)
        }
        fn seq<T: Parse>(&mut self) -> Seq<T, Self, DefaultParser> {
            self.seq_with(DefaultParser)
        }
        fn seq_with<T, P: Parser<T>>(&mut self, parser: P) -> Seq<T, Self, P> {
            Seq {
                input: self,
                parser,
                marker: PhantomData,
            }
        }
        fn collect<T: Parse, C: std::iter::FromIterator<T>>(&mut self, n: usize) -> C {
            self.seq().take(n).collect()
        }
    }
    impl<T: Input> Input for &mut T {
        fn bytes(&mut self) -> &[u8] {
            (**self).bytes()
        }
    }
    pub trait Parser<T> {
        fn parse<I: Input + ?Sized>(&mut self, s: &mut I) -> T;
    }
    impl<T, P: Parser<T>> Parser<T> for &mut P {
        fn parse<I: Input + ?Sized>(&mut self, s: &mut I) -> T {
            (**self).parse(s)
        }
    }
    pub trait Parse {
        fn parse<I: Input + ?Sized>(s: &mut I) -> Self;
    }
    pub struct DefaultParser;
    impl<T: Parse> Parser<T> for DefaultParser {
        fn parse<I: Input + ?Sized>(&mut self, s: &mut I) -> T {
            T::parse(s)
        }
    }
    pub struct Seq<'a, T, I: ?Sized, P> {
        input: &'a mut I,
        parser: P,
        marker: PhantomData<*const T>,
    }
    impl<'a, T, I: Input + ?Sized, P: Parser<T>> Iterator for Seq<'a, T, I, P> {
        type Item = T;
        #[inline]
        fn next(&mut self) -> Option<Self::Item> {
            Some(self.input.parse_with(&mut self.parser))
        }
        fn size_hint(&self) -> (usize, Option<usize>) {
            (!0, None)
        }
    }
    impl Parse for char {
        #[inline]
        fn parse<I: Input + ?Sized>(s: &mut I) -> Self {
            let s = s.bytes();
            debug_assert_eq!(s.len(), 1);
            *s.first().expect("zero length") as char
        }
    }
    macro_rules ! tuple { ($ ($ T : ident) ,*) => { impl <$ ($ T : Parse) ,*> Parse for ($ ($ T ,) *) { # [inline] # [allow (unused_variables)] # [allow (clippy :: unused_unit)] fn parse < I : Input + ? Sized > (s : & mut I) -> Self { ($ ($ T :: parse (s) ,) *) } } } ; }
    tuple!();
    tuple!(A);
    tuple!(A, B);
    tuple!(A, B, C);
    tuple!(A, B, C, D);
    tuple!(A, B, C, D, E);
    tuple!(A, B, C, D, E, F);
    tuple!(A, B, C, D, E, F, G);
    #[cfg(feature = "newer")]
    impl<T: Parse, const N: usize> Parse for [T; N] {
        fn parse<I: Input + ?Sized>(s: &mut I) -> Self {
            use std::{
                mem::{self, MaybeUninit},
                ptr,
            };
            struct Guard<T, const N: usize> {
                arr: [MaybeUninit<T>; N],
                i: usize,
            }
            impl<T, const N: usize> Drop for Guard<T, N> {
                fn drop(&mut self) {
                    unsafe {
                        ptr::drop_in_place(&mut self.arr[..self.i] as *mut _ as *mut [T]);
                    }
                }
            }
            let mut g = Guard::<T, N> {
                arr: unsafe { MaybeUninit::uninit().assume_init() },
                i: 0,
            };
            while g.i < N {
                g.arr[g.i] = MaybeUninit::new(s.parse());
                g.i += 1;
            }
            unsafe { mem::transmute_copy(&g.arr) }
        }
    }
    macro_rules! uint {
        ($ ty : ty) => {
            impl Parse for $ty {
                #[inline]
                fn parse<I: Input + ?Sized>(s: &mut I) -> Self {
                    let s = s.bytes();
                    s.iter().fold(0, |x, d| 10 * x + (0xf & d) as $ty)
                }
            }
        };
    }
    macro_rules! int {
        ($ ty : ty) => {
            impl Parse for $ty {
                #[inline]
                fn parse<I: Input + ?Sized>(s: &mut I) -> Self {
                    let f = |s: &[u8]| {
                        s.iter()
                            .fold(0 as $ty, |x, d| (10 * x).wrapping_add((0xf & d) as $ty))
                    };
                    let s = s.bytes();
                    if let Some((b'-', s)) = s.split_first() {
                        f(s).wrapping_neg()
                    } else {
                        f(s)
                    }
                }
            }
        };
    }
    macro_rules! float {
        ($ ty : ty) => {
            impl Parse for $ty {
                fn parse<I: Input + ?Sized>(s: &mut I) -> Self {
                    const POW: [$ty; 18] = [
                        1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11, 1e12, 1e13,
                        1e14, 1e15, 1e16, 1e17,
                    ];
                    let s = s.bytes();
                    let (minus, s) = if let Some((b'-', s)) = s.split_first() {
                        (true, s)
                    } else {
                        (false, s)
                    };
                    let (int, fract) = if let Some(p) = s.iter().position(|c| *c == b'.') {
                        (&s[..p], &s[p + 1..])
                    } else {
                        (s, &[][..])
                    };
                    let x = int
                        .iter()
                        .chain(fract)
                        .fold(0u64, |x, d| 10 * x + (0xf & *d) as u64);
                    let x = x as $ty;
                    let x = if minus { -x } else { x };
                    let exp = fract.len();
                    if exp == 0 {
                        x
                    } else if let Some(pow) = POW.get(exp) {
                        x / pow
                    } else {
                        x / (10.0 as $ty).powi(exp as i32)
                    }
                }
            }
        };
    }
    macro_rules! from_bytes {
        ($ ty : ty) => {
            impl Parse for $ty {
                #[inline]
                fn parse<I: Input + ?Sized>(s: &mut I) -> Self {
                    s.bytes().into()
                }
            }
        };
    }
    macro_rules! from_str {
        ($ ty : ty) => {
            impl Parse for $ty {
                #[inline]
                fn parse<I: Input + ?Sized>(s: &mut I) -> Self {
                    s.str().into()
                }
            }
        };
    }
    macro_rules ! impls { ($ m : ident , $ ($ ty : ty) ,*) => { $ ($ m ! ($ ty) ;) * } ; }
    impls!(uint, usize, u8, u16, u32, u64, u128);
    impls!(int, isize, i8, i16, i32, i64, i128);
    impls!(float, f32, f64);
    impls!(from_bytes, Vec<u8>, Box<[u8]>);
    impls!(from_str, String);
    #[derive(Clone)]
    pub struct SplitWs<T> {
        src: T,
        buf: Vec<u8>,
        pos: usize,
        len: usize,
    }
    const BUF_SIZE: usize = 1 << 26;
    impl<T: Read> SplitWs<T> {
        pub fn new(src: T) -> Self {
            Self {
                src,
                buf: vec![0; BUF_SIZE],
                pos: 0,
                len: 0,
            }
        }
        #[inline(always)]
        fn peek(&self) -> &[u8] {
            unsafe { self.buf.get_unchecked(self.pos..self.len) }
        }
        #[inline(always)]
        fn consume(&mut self, n: usize) -> &[u8] {
            let pos = self.pos;
            self.pos += n;
            unsafe { self.buf.get_unchecked(pos..self.pos) }
        }
        fn read(&mut self) -> usize {
            self.buf.copy_within(self.pos..self.len, 0);
            self.len -= self.pos;
            self.pos = 0;
            if self.len == self.buf.len() {
                self.buf.resize(2 * self.buf.len(), 0);
            }
            loop {
                match self.src.read(&mut self.buf[self.len..]) {
                    Ok(n) => {
                        self.len += n;
                        return n;
                    }
                    Err(e) if e.kind() == io::ErrorKind::WouldBlock => {}
                    Err(e) => panic!("io error: {:?}", e),
                }
            }
        }
    }
    impl<T: Read> Input for SplitWs<T> {
        #[inline]
        fn bytes(&mut self) -> &[u8] {
            loop {
                if let Some(del) = self.peek().iter().position(|c| c.is_ascii_whitespace()) {
                    if del > 0 {
                        let s = self.consume(del + 1);
                        return s.split_last().unwrap().1;
                    } else {
                        self.consume(1);
                    }
                } else if self.read() == 0 {
                    return self.consume(self.len - self.pos);
                }
            }
        }
    }
}
pub mod macros {
    #[macro_export]
    macro_rules ! w { ($ ($ arg : tt) *) => { write ! ($ ($ arg) *) . unwrap () ; } }
    #[macro_export]
    macro_rules ! wln { ($ dst : expr $ (, $ ($ arg : tt) *) ?) => { { writeln ! ($ dst $ (, $ ($ arg) *) ?) . unwrap () ; # [cfg (debug_assertions)] $ dst . flush () . unwrap () ; } } }
    #[macro_export]
    macro_rules! w_iter {
        ($ dst : expr , $ fmt : expr , $ iter : expr , $ delim : expr) => {{
            let mut first = true;
            for elem in $iter {
                if first {
                    w!($dst, $fmt, elem);
                    first = false;
                } else {
                    w!($dst, concat!($delim, $fmt), elem);
                }
            }
        }};
        ($ dst : expr , $ fmt : expr , $ iter : expr) => {
            w_iter!($dst, $fmt, $iter, " ")
        };
    }
    #[macro_export]
    macro_rules ! w_iter_ln { ($ dst : expr , $ ($ t : tt) *) => { { w_iter ! ($ dst , $ ($ t) *) ; wln ! ($ dst) ; } } }
    #[macro_export]
    macro_rules ! e { ($ ($ t : tt) *) => { # [cfg (debug_assertions)] eprint ! ($ ($ t) *) } }
    #[macro_export]
    macro_rules ! eln { ($ ($ t : tt) *) => { # [cfg (debug_assertions)] eprintln ! ($ ($ t) *) } }
    #[macro_export]
    #[doc(hidden)]
    macro_rules ! __tstr { ($ h : expr $ (, $ t : expr) +) => { concat ! (__tstr ! ($ ($ t) ,+) , ", " , __tstr ! (@)) } ; ($ h : expr) => { concat ! (__tstr ! () , " " , __tstr ! (@)) } ; () => { "\x1B[94m[{}:{}]\x1B[0m" } ; (@) => { "\x1B[1;92m{}\x1B[0m = {:?}" } }
    #[macro_export]
    macro_rules ! d { ($ ($ a : expr) ,*) => { if std :: env :: var ("ND") . map (| v | & v == "0") . unwrap_or (true) { eln ! (__tstr ! ($ ($ a) ,*) , file ! () , line ! () , $ (stringify ! ($ a) , $ a) ,*) ; } } ; }
}