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namespace ModInt {

using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;

template <class T> constexpr T power(T a, u64 b, T res = 1) {
    for (; b != 0; b /= 2, a *= a) {
        if (b & 1) {
            res *= a;
        }
    }
    return res;
}

template <u32 P> constexpr u32 mulMod(u32 a, u32 b) { return u64(a) * b % P; }

template <u64 P> constexpr u64 mulMod(u64 a, u64 b) {
    u64 res = a * b - u64(1.L * a * b / P - 0.5L) * P;
    res %= P;
    return res;
}

constexpr i64 safeMod(i64 x, i64 m) {
    x %= m;
    if (x < 0) {
        x += m;
    }
    return x;
}

constexpr std::pair<i64, i64> invGcd(i64 a, i64 b) {
    a = safeMod(a, b);
    if (a == 0) {
        return {b, 0};
    }

    i64 s = b, t = a;
    i64 m0 = 0, m1 = 1;

    while (t) {
        i64 u = s / t;
        s -= t * u;
        m0 -= m1 * u;

        std::swap(s, t);
        std::swap(m0, m1);
    }

    if (m0 < 0) {
        m0 += b / s;
    }

    return {s, m0};
}

template <std::unsigned_integral U, U P> struct ModIntBase {
  public:
    constexpr ModIntBase() : x(0) {}
    template <std::unsigned_integral T> constexpr ModIntBase(T x_) : x(x_ % mod()) {}
    template <std::signed_integral T> constexpr ModIntBase(T x_) {
        using S = std::make_signed_t<U>;
        S v = x_ % S(mod());
        if (v < 0) {
            v += mod();
        }
        x = v;
    }

    constexpr static U mod() { return P; }

    constexpr U val() const { return x; }

    constexpr ModIntBase operator-() const {
        ModIntBase res;
        res.x = (x == 0 ? 0 : mod() - x);
        return res;
    }

    constexpr ModIntBase inv() const { return power(*this, mod() - 2); }

    constexpr ModIntBase &operator*=(const ModIntBase &rhs) & {
        x = mulMod<mod()>(x, rhs.val());
        return *this;
    }
    constexpr ModIntBase &operator+=(const ModIntBase &rhs) & {
        x += rhs.val();
        if (x >= mod()) {
            x -= mod();
        }
        return *this;
    }
    constexpr ModIntBase &operator-=(const ModIntBase &rhs) & {
        x -= rhs.val();
        if (x >= mod()) {
            x += mod();
        }
        return *this;
    }
    constexpr ModIntBase &operator/=(const ModIntBase &rhs) & { return *this *= rhs.inv(); }

    friend constexpr ModIntBase operator*(ModIntBase lhs, const ModIntBase &rhs) {
        lhs *= rhs;
        return lhs;
    }
    friend constexpr ModIntBase operator+(ModIntBase lhs, const ModIntBase &rhs) {
        lhs += rhs;
        return lhs;
    }
    friend constexpr ModIntBase operator-(ModIntBase lhs, const ModIntBase &rhs) {
        lhs -= rhs;
        return lhs;
    }
    friend constexpr ModIntBase operator/(ModIntBase lhs, const ModIntBase &rhs) {
        lhs /= rhs;
        return lhs;
    }

    friend constexpr std::istream &operator>>(std::istream &is, ModIntBase &a) {
        i64 i;
        is >> i;
        a = i;
        return is;
    }
    friend constexpr std::ostream &operator<<(std::ostream &os, const ModIntBase &a) {
        return os << a.val();
    }

    friend constexpr bool operator==(const ModIntBase &lhs, const ModIntBase &rhs) {
        return lhs.val() == rhs.val();
    }
    friend constexpr std::strong_ordering operator<=>(const ModIntBase &lhs,
                                                      const ModIntBase &rhs) {
        return lhs.val() <=> rhs.val();
    }

  private:
    U x;
};

template <u32 P> using ModInt = ModIntBase<u32, P>;
template <u64 P> using ModInt64 = ModIntBase<u64, P>;

} // namespace ModInt