proconlib

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:heavy_check_mark: Automatically deduced multiplicative structure (tools/multiplicative_structure.hpp) 

template <typename T>
using multiplicative_structure = /* tools::monoids::multiplies<T> or tools::groups::multiplies<T> */;

It automatically selects either tools::monoids::multiplies<T> or tools::groups::multiplies<T> based on T.

Example

T tools::multiplicative_structure<T>
int tools::monoids::multiplies<int>
double tools::groups::multiplies<double>
std::complex<double> tools::groups::multiplies<std::complex<double>>
atcoder::modint998244353 tools::groups::multiplies<atcoder::modint998244353>
atcoder::static_modint<4> tools::monoids::multiplies<atcoder::static_modint<4>>
atcoder::modint tools::groups::multiplies<atcoder::modint>
tools::fps<atcoder::modint998244353> tools::groups::multiplies<tools::fps<atcoder::modint998244353>>
tools::permutation<int> tools::monoids::multiplies<tools::permutation<int>>

Constraints

Time Complexity

License

Author

Depends on

Required by

Verified with

Code

#ifndef TOOLS_MULTIPLICATIVE_STRUCTURE_HPP
#define TOOLS_MULTIPLICATIVE_STRUCTURE_HPP

#include <concepts>
#include <type_traits>
#include "atcoder/modint.hpp"
#include "tools/complex.hpp"
#include "tools/groups.hpp"
#include "tools/integral.hpp"
#include "tools/is_prime.hpp"
#include "tools/monoids.hpp"
#include "tools/prime_static_modint.hpp"

namespace tools {
  template <typename T>
  using multiplicative_structure = std::conditional_t<
    tools::complex<T> || std::floating_point<T> || tools::prime_static_modint<T> || atcoder::internal::is_dynamic_modint<T>::value,
      tools::groups::multiplies<T>,
      std::conditional_t<
        tools::integral<T> || atcoder::internal::is_static_modint<T>::value,
          tools::monoids::multiplies<T>,
          std::conditional_t<
            requires(T a, T b) { { a / b } -> std::same_as<T>; },
              tools::groups::multiplies<T>,
              tools::monoids::multiplies<T>
          >
      >
  >;
}

#endif

#line 1 "tools/multiplicative_structure.hpp"



#include <concepts>
#include <type_traits>
#line 1 "lib/ac-library/atcoder/modint.hpp"



#include <cassert>
#include <numeric>
#line 7 "lib/ac-library/atcoder/modint.hpp"

#ifdef _MSC_VER
#include <intrin.h>
#endif

#line 1 "lib/ac-library/atcoder/internal_math.hpp"



#include <utility>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m`
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [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
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned long long y = x * _m;
        return (unsigned int)(z - y + (z < y ? _m : 0));
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    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
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long 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

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

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        // y_max < m * (n + 1)
        // floor(y_max / m) <= n
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder


#line 1 "lib/ac-library/atcoder/internal_type_traits.hpp"



#line 7 "lib/ac-library/atcoder/internal_type_traits.hpp"

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder


#line 14 "lib/ac-library/atcoder/modint.hpp"

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

    mint operator+() const { return *this; }
    mint operator-() const { return mint() - *this; }

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder


#line 1 "tools/complex.hpp"



#include <complex>
#line 1 "tools/specialization_of.hpp"



#line 5 "tools/specialization_of.hpp"

namespace tools {
  namespace detail {
    namespace specialization_of {
      template <typename, template <typename...> typename>
      struct trait : std::false_type {};

      template <template <typename...> typename U, typename... Args>
      struct trait<U<Args...>, U> : std::true_type {};
    }
  }

  template <typename T, template <typename...> typename U>
  concept specialization_of = tools::detail::specialization_of::trait<T, U>::value;
}


#line 6 "tools/complex.hpp"

namespace tools {
  template <typename T>
  concept complex = tools::specialization_of<T, std::complex>;
}


#line 1 "tools/groups.hpp"



#include <cstddef>
#line 1 "tools/arithmetic.hpp"



#line 1 "tools/integral.hpp"



#line 1 "tools/is_integral.hpp"



#line 5 "tools/is_integral.hpp"

namespace tools {
  template <typename T>
  struct is_integral : std::is_integral<T> {};

  template <typename T>
  inline constexpr bool is_integral_v = tools::is_integral<T>::value;
}


#line 5 "tools/integral.hpp"

namespace tools {
  template <typename T>
  concept integral = tools::is_integral_v<T>;
}


#line 6 "tools/arithmetic.hpp"

namespace tools {
  template <typename T>
  concept arithmetic = tools::integral<T> || std::floating_point<T>;
}


#line 7 "tools/groups.hpp"

namespace tools {
  namespace groups {
    template <typename G>
    struct bit_xor {
      using T = G;
      static T op(const T& x, const T& y) {
        return x ^ y;
      }
      static T e() {
        return T(0);
      }
      static T inv(const T& x) {
        return x;
      }
    };

    template <typename G>
    struct multiplies {
      using T = G;
      static T op(const T& x, const T& y) {
        return x * y;
      }
      static T e() {
        return T(1);
      }
      static T inv(const T& x) {
        return e() / x;
      }
    };

    template <typename G>
    struct plus {
      using T = G;
      static T op(const T& x, const T& y) {
        return x + y;
      }
      static T e() {
        return T(0);
      }
      static T inv(const T& x) {
        return -x;
      }
    };
  }
}


#line 1 "tools/is_prime.hpp"



#include <array>
#line 1 "tools/prod_mod.hpp"



#line 1 "tools/uint128_t.hpp"



#line 1 "tools/detail/int128_t_and_uint128_t.hpp"



#include <algorithm>
#line 7 "tools/detail/int128_t_and_uint128_t.hpp"
#include <cstdint>
#include <functional>
#include <iostream>
#include <limits>
#include <string>
#include <string_view>
#line 1 "tools/abs.hpp"



#include <cmath>
#line 7 "tools/abs.hpp"

namespace tools {
  namespace detail::abs {
    template <typename T>
    struct impl {
      constexpr decltype(auto) operator()(const T x) const noexcept(noexcept(std::abs(x))) {
        return std::abs(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) abs(T&& x) noexcept(noexcept(tools::detail::abs::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::abs::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/bit_ceil.hpp"



#include <bit>
#line 1 "tools/is_signed.hpp"



#line 5 "tools/is_signed.hpp"

namespace tools {
  template <typename T>
  struct is_signed : std::is_signed<T> {};

  template <typename T>
  inline constexpr bool is_signed_v = tools::is_signed<T>::value;
}


#line 1 "tools/is_unsigned.hpp"



#line 5 "tools/is_unsigned.hpp"

namespace tools {
  template <typename T>
  struct is_unsigned : std::is_unsigned<T> {};

  template <typename T>
  inline constexpr bool is_unsigned_v = tools::is_unsigned<T>::value;
}


#line 1 "tools/make_unsigned.hpp"



#line 5 "tools/make_unsigned.hpp"

namespace tools {
  template <typename T>
  struct make_unsigned : std::make_unsigned<T> {};

  template <typename T>
  using make_unsigned_t = typename tools::make_unsigned<T>::type;
}


#line 1 "tools/non_bool_integral.hpp"



#line 7 "tools/non_bool_integral.hpp"

namespace tools {
  template <typename T>
  concept non_bool_integral = tools::integral<T> && !std::same_as<std::remove_cv_t<T>, bool>;
}


#line 12 "tools/bit_ceil.hpp"

namespace tools {
  namespace detail::bit_ceil {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr T operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr T operator()(const T x) const noexcept(noexcept(std::bit_ceil(x))) requires tools::is_unsigned_v<T> {
        return std::bit_ceil(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) bit_ceil(T&& x) noexcept(noexcept(tools::detail::bit_ceil::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::bit_ceil::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/bit_floor.hpp"



#line 12 "tools/bit_floor.hpp"

namespace tools {
  namespace detail::bit_floor {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr T operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr T operator()(const T x) const noexcept(noexcept(std::bit_floor(x))) requires tools::is_unsigned_v<T> {
        return std::bit_floor(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) bit_floor(T&& x) noexcept(noexcept(tools::detail::bit_floor::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::bit_floor::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/bit_width.hpp"



#line 12 "tools/bit_width.hpp"

namespace tools {
  namespace detail::bit_width {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr int operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr int operator()(const T x) const noexcept(noexcept(std::bit_width(x))) requires tools::is_unsigned_v<T> {
        return std::bit_width(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) bit_width(T&& x) noexcept(noexcept(tools::detail::bit_width::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::bit_width::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/countr_zero.hpp"



#line 14 "tools/countr_zero.hpp"

namespace tools {
  namespace detail::countr_zero {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr int operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return std::min(impl<tools::make_unsigned_t<T>>{}(x), std::numeric_limits<T>::digits);
      }
      constexpr int operator()(const T x) const noexcept(noexcept(std::countr_zero(x))) requires tools::is_unsigned_v<T> {
        return std::countr_zero(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) countr_zero(T&& x) noexcept(noexcept(tools::detail::countr_zero::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::countr_zero::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/gcd.hpp"



#line 7 "tools/gcd.hpp"

namespace tools {
  namespace detail::gcd {
    template <typename M, typename N>
    struct impl {
      constexpr decltype(auto) operator()(const M m, const N n) const noexcept(noexcept(std::gcd(m, n))) {
        return std::gcd(m, n);
      }
    };
  }

  template <typename M, typename N>
  constexpr decltype(auto) gcd(M&& m, N&& n) noexcept(noexcept(tools::detail::gcd::impl<std::remove_cvref_t<M>, std::remove_cvref_t<N>>{}(std::forward<M>(m), std::forward<N>(n)))) {
    return tools::detail::gcd::impl<std::remove_cvref_t<M>, std::remove_cvref_t<N>>{}(std::forward<M>(m), std::forward<N>(n));
  }
}


#line 1 "tools/has_single_bit.hpp"



#line 12 "tools/has_single_bit.hpp"

namespace tools {
  namespace detail::has_single_bit {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr bool operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr bool operator()(const T x) const noexcept(noexcept(std::has_single_bit(x))) requires tools::is_unsigned_v<T> {
        return std::has_single_bit(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) has_single_bit(T&& x) noexcept(noexcept(tools::detail::has_single_bit::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::has_single_bit::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/hash_combine.hpp"



#line 6 "tools/hash_combine.hpp"

// Source: https://github.com/google/cityhash/blob/f5dc54147fcce12cefd16548c8e760d68ac04226/src/city.h
// License: MIT
// Author: Google Inc.

// Copyright (c) 2011 Google, Inc.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.

namespace tools {
  template <typename T>
  void hash_combine(std::size_t& seed, const T& v) {
    static const std::hash<T> hasher;
    static constexpr std::size_t k_mul = 0x9ddfea08eb382d69ULL;
    std::size_t a = (hasher(v) ^ seed) * k_mul;
    a ^= (a >> 47);
    std::size_t b = (seed ^ a) * k_mul;
    b ^= (b >> 47);
    seed = b * k_mul;
  }
}


#line 1 "tools/make_signed.hpp"



#line 5 "tools/make_signed.hpp"

namespace tools {
  template <typename T>
  struct make_signed : std::make_signed<T> {};

  template <typename T>
  using make_signed_t = typename tools::make_signed<T>::type;
}


#line 1 "tools/now.hpp"



#include <chrono>

namespace tools {
  inline long long now() {
    return std::chrono::duration_cast<std::chrono::nanoseconds>(std::chrono::high_resolution_clock::now().time_since_epoch()).count();
  }
}


#line 1 "tools/popcount.hpp"



#line 12 "tools/popcount.hpp"

namespace tools {
  namespace detail::popcount {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr int operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr int operator()(const T x) const noexcept(noexcept(std::popcount(x))) requires tools::is_unsigned_v<T> {
        return std::popcount(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) popcount(T&& x) noexcept(noexcept(tools::detail::popcount::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::popcount::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 31 "tools/detail/int128_t_and_uint128_t.hpp"

namespace tools {
  using uint128_t = unsigned __int128;
  using int128_t = __int128;

  template <>
  struct is_integral<tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<tools::uint128_t> : std::true_type {};
  template <>
  struct is_integral<const tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<const tools::uint128_t> : std::true_type {};
  template <>
  struct is_integral<volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<volatile tools::uint128_t> : std::true_type {};
  template <>
  struct is_integral<const volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<const volatile tools::uint128_t> : std::true_type {};

  template <>
  struct is_signed<tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<tools::uint128_t> : std::false_type {};
  template <>
  struct is_signed<const tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<const tools::uint128_t> : std::false_type {};
  template <>
  struct is_signed<volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<volatile tools::uint128_t> : std::false_type {};
  template <>
  struct is_signed<const volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<const volatile tools::uint128_t> : std::false_type {};

  template <>
  struct is_unsigned<tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<tools::uint128_t> : std::true_type {};
  template <>
  struct is_unsigned<const tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<const tools::uint128_t> : std::true_type {};
  template <>
  struct is_unsigned<volatile tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<volatile tools::uint128_t> : std::true_type {};
  template <>
  struct is_unsigned<const volatile tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<const volatile tools::uint128_t> : std::true_type {};

  template <>
  struct make_signed<tools::int128_t> {
    using type = tools::int128_t;
  };
  template <>
  struct make_signed<tools::uint128_t> {
    using type = tools::int128_t;
  };
  template <>
  struct make_signed<const tools::int128_t> {
    using type = const tools::int128_t;
  };
  template <>
  struct make_signed<const tools::uint128_t> {
    using type = const tools::int128_t;
  };
  template <>
  struct make_signed<volatile tools::int128_t> {
    using type = volatile tools::int128_t;
  };
  template <>
  struct make_signed<volatile tools::uint128_t> {
    using type = volatile tools::int128_t;
  };
  template <>
  struct make_signed<const volatile tools::int128_t> {
    using type = const volatile tools::int128_t;
  };
  template <>
  struct make_signed<const volatile tools::uint128_t> {
    using type = const volatile tools::int128_t;
  };

  template <>
  struct make_unsigned<tools::int128_t> {
    using type = tools::uint128_t;
  };
  template <>
  struct make_unsigned<tools::uint128_t> {
    using type = tools::uint128_t;
  };
  template <>
  struct make_unsigned<const tools::int128_t> {
    using type = const tools::uint128_t;
  };
  template <>
  struct make_unsigned<const tools::uint128_t> {
    using type = const tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile tools::int128_t> {
    using type = volatile tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile tools::uint128_t> {
    using type = volatile tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile tools::int128_t> {
    using type = const volatile tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile tools::uint128_t> {
    using type = const volatile tools::uint128_t;
  };

  namespace detail::int128_t {
    constexpr tools::uint128_t parse_unsigned(const std::string_view s) noexcept {
      assert(!s.empty());
      tools::uint128_t x = 0;
      std::size_t i = s[0] == '+';
      if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
        for (i += 2; i < s.size(); ++i) {
          assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
          x <<= 4;
          if ('0' <= s[i] && s[i] <= '9') {
            x |= s[i] - '0';
          } else if ('a' <= s[i] && s[i] <= 'f') {
            x |= s[i] - 'a' + 10;
          } else {
            x |= s[i] - 'A' + 10;
          }
        }
      } else {
        for (; i < s.size(); ++i) {
          assert('0' <= s[i] && s[i] <= '9');
          x *= 10;
          x += s[i] - '0';
        }
      }
      return x;
    }

    constexpr tools::int128_t parse_signed(const std::string_view s) noexcept {
      assert(!s.empty());
      tools::int128_t x = 0;
      if (s[0] == '-') {
        std::size_t i = 1;
        if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
          for (i += 2; i < s.size(); ++i) {
            assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
            x *= 16;
            if ('0' <= s[i] && s[i] <= '9') {
              x -= s[i] - '0';
            } else if ('a' <= s[i] && s[i] <= 'f') {
              x -= s[i] - 'a' + 10;
            } else {
              x -= s[i] - 'A' + 10;
            }
          }
        } else {
          for (; i < s.size(); ++i) {
            assert('0' <= s[i] && s[i] <= '9');
            x *= 10;
            x -= s[i] - '0';
          }
        }
      } else {
        std::size_t i = s[0] == '+';
        if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
          for (i += 2; i < s.size(); ++i) {
            assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
            x <<= 4;
            if ('0' <= s[i] && s[i] <= '9') {
              x |= s[i] - '0';
            } else if ('a' <= s[i] && s[i] <= 'f') {
              x |= s[i] - 'a' + 10;
            } else {
              x |= s[i] - 'A' + 10;
            }
          }
        } else {
          for (; i < s.size(); ++i) {
            assert('0' <= s[i] && s[i] <= '9');
            x *= 10;
            x += s[i] - '0';
          }
        }
      }
      return x;
    }
  }
}

#define UINT128_C(c) tools::detail::int128_t::parse_unsigned(#c)
#define INT128_C(c) tools::detail::int128_t::parse_signed(#c)

inline std::istream& operator>>(std::istream& is, tools::uint128_t& x) {
  std::string s;
  is >> s;
  x = tools::detail::int128_t::parse_unsigned(s);
  return is;
}
inline std::istream& operator>>(std::istream& is, tools::int128_t& x) {
  std::string s;
  is >> s;
  x = tools::detail::int128_t::parse_signed(s);
  return is;
}

inline std::ostream& operator<<(std::ostream& os, tools::uint128_t x) {
  std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else {
    s.push_back('0');
  }

  std::ranges::reverse(s);
  return os << s;
}
inline std::ostream& operator<<(std::ostream& os, tools::int128_t x) {
  std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else if (x < 0) {
    while (x < 0) {
      s.push_back('0' + (-(x % 10)));
      x /= 10;
    }
    s.push_back('-');
  } else {
    s.push_back('0');
  }

  std::ranges::reverse(s);
  return os << s;
}

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
namespace std {
  template <>
  struct hash<tools::uint128_t> {
    std::size_t operator()(const tools::uint128_t& x) const {
      static const std::size_t seed = tools::now();

      std::size_t hash = seed;
      tools::hash_combine(hash, static_cast<std::uint64_t>(x >> 64));
      tools::hash_combine(hash, static_cast<std::uint64_t>(x & ((UINT128_C(1) << 64) - 1)));
      return hash;
    }
  };
  template <>
  struct hash<tools::int128_t> {
    std::size_t operator()(const tools::int128_t& x) const {
      static std::hash<tools::uint128_t> hasher;
      return hasher(static_cast<tools::uint128_t>(x));
    }
  };
}
#endif

namespace tools {
  template <>
  struct detail::abs::impl<tools::int128_t> {
    constexpr tools::int128_t operator()(const tools::int128_t& x) const noexcept {
      return x >= 0 ? x : -x;
    }
  };

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
  template <>
  struct detail::bit_ceil::impl<tools::uint128_t> {
    constexpr tools::uint128_t operator()(tools::uint128_t x) const noexcept {
      if (x <= 1) return 1;
      --x;
      x |= x >> 1;
      x |= x >> 2;
      x |= x >> 4;
      x |= x >> 8;
      x |= x >> 16;
      x |= x >> 32;
      x |= x >> 64;
      return ++x;
    }
  };

  template <>
  struct detail::bit_floor::impl<tools::uint128_t> {
    constexpr tools::uint128_t operator()(tools::uint128_t x) const noexcept {
      x |= x >> 1;
      x |= x >> 2;
      x |= x >> 4;
      x |= x >> 8;
      x |= x >> 16;
      x |= x >> 32;
      x |= x >> 64;
      return x & ~(x >> 1);
    }
  };

  template <>
  struct detail::bit_width::impl<tools::uint128_t> {
    constexpr int operator()(tools::uint128_t x) const noexcept {
      int w = 0;
      if (x & UINT128_C(0xffffffffffffffff0000000000000000)) {
        x >>= 64;
        w += 64;
      }
      if (x & UINT128_C(0xffffffff00000000)) {
        x >>= 32;
        w += 32;
      }
      if (x & UINT128_C(0xffff0000)) {
        x >>= 16;
        w += 16;
      }
      if (x & UINT128_C(0xff00)) {
        x >>= 8;
        w += 8;
      }
      if (x & UINT128_C(0xf0)) {
        x >>= 4;
        w += 4;
      }
      if (x & UINT128_C(0xc)) {
        x >>= 2;
        w += 2;
      }
      if (x & UINT128_C(0x2)) {
        x >>= 1;
        w += 1;
      }
      w += x;
      return w;
    }
  };

  template <>
  class detail::countr_zero::impl<tools::uint128_t> {
    using type = tools::uint128_t;
    static constexpr int shift = 120;
    static constexpr type magic = UINT128_C(0x01061438916347932a5cd9d3ead7b77f);
    static constexpr int ntz_table[255] = {
      128,   0,   1,  -1,   2,  -1,   8,  -1,   3,  -1,  15,  -1,   9,  -1,  22,  -1,
        4,  -1,  29,  -1,  16,  -1,  36,  -1,  10,  -1,  43,  -1,  23,  -1,  50,  -1,
        5,  -1,  33,  -1,  30,  -1,  57,  -1,  17,  -1,  64,  -1,  37,  -1,  71,  -1,
       11,  -1,  60,  -1,  44,  -1,  78,  -1,  24,  -1,  85,  -1,  51,  -1,  92,  -1,
       -1,   6,  -1,  20,  -1,  34,  -1,  48,  31,  -1,  -1,  69,  58,  -1,  -1,  90,
       18,  -1,  67,  -1,  65,  -1,  99,  -1,  38,  -1, 101,  -1,  72,  -1, 106,  -1,
       -1,  12,  -1,  40,  -1,  61,  -1,  82,  45,  -1,  -1, 103,  79,  -1, 113,  -1,
       -1,  25,  -1,  74,  86,  -1,  -1, 116,  -1,  52,  -1, 108,  -1,  93,  -1, 120,
      127,  -1,  -1,   7,  -1,  14,  -1,  21,  -1,  28,  -1,  35,  -1,  42,  -1,  49,
       -1,  32,  -1,  56,  -1,  63,  -1,  70,  -1,  59,  -1,  77,  -1,  84,  -1,  91,
       -1,  19,  -1,  47,  -1,  68,  -1,  89,  -1,  66,  -1,  98,  -1, 100,  -1, 105,
       -1,  39,  -1,  81,  -1, 102,  -1, 112,  -1,  73,  -1, 115,  -1, 107,  -1, 119,
      126,  -1,  13,  -1,  27,  -1,  41,  -1,  -1,  55,  62,  -1,  -1,  76,  83,  -1,
       -1,  46,  -1,  88,  -1,  97,  -1, 104,  -1,  80,  -1, 111,  -1, 114,  -1, 118,
      125,  -1,  26,  -1,  54,  -1,  75,  -1,  -1,  87,  96,  -1,  -1, 110,  -1, 117,
      124,  -1,  53,  -1,  -1,  95, 109,  -1, 123,  -1,  94,  -1, 122,  -1, 121
    };

  public:
    constexpr int operator()(const type& x) const noexcept {
      return ntz_table[static_cast<type>(magic * static_cast<type>(x & -x)) >> shift];
    }
  };

  namespace detail::gcd {
    template <>
    struct impl<tools::uint128_t, tools::uint128_t> {
      constexpr tools::uint128_t operator()(tools::uint128_t m, tools::uint128_t n) const noexcept {
        while (n != 0) {
          m %= n;
          std::swap(m, n);
        }
        return m;
      };
    };

    template <typename T>
    concept non_bool_integral_at_most_128bit = tools::non_bool_integral<T> && std::numeric_limits<T>::digits <= 128;
    template <typename T>
    concept non_bool_integral_at_most_64bit = tools::non_bool_integral<T> && std::numeric_limits<T>::digits <= 64;

    template <typename M, typename N> requires (
      (non_bool_integral_at_most_128bit<M> && non_bool_integral_at_most_128bit<N>)
      && !(non_bool_integral_at_most_64bit<M> && non_bool_integral_at_most_64bit<N>)
      && !(std::same_as<M, tools::uint128_t> && std::same_as<N, tools::uint128_t>)
    )
    struct impl<M, N> {
      constexpr std::common_type_t<M, N> operator()(const M m, const N n) const noexcept {
        return std::common_type_t<M, N>(
          tools::gcd(
            m >= 0 ? tools::uint128_t(m) : tools::uint128_t(-(m + 1)) + 1,
            n >= 0 ? tools::uint128_t(n) : tools::uint128_t(-(n + 1)) + 1
          )
        );
      }
    };
  }

  template <>
  struct detail::has_single_bit::impl<tools::uint128_t> {
    constexpr bool operator()(tools::uint128_t x) const noexcept {
      return x != 0 && (x & (x - 1)) == 0;
    }
  };

  template <>
  struct detail::popcount::impl<tools::uint128_t> {
    constexpr int operator()(tools::uint128_t x) const noexcept {
      x = (x & UINT128_C(0x55555555555555555555555555555555)) + (x >> 1 & UINT128_C(0x55555555555555555555555555555555));
      x = (x & UINT128_C(0x33333333333333333333333333333333)) + (x >> 2 & UINT128_C(0x33333333333333333333333333333333));
      x = (x & UINT128_C(0x0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f)) + (x >> 4 & UINT128_C(0x0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f));
      x = (x & UINT128_C(0x00ff00ff00ff00ff00ff00ff00ff00ff)) + (x >> 8 & UINT128_C(0x00ff00ff00ff00ff00ff00ff00ff00ff));
      x = (x & UINT128_C(0x0000ffff0000ffff0000ffff0000ffff)) + (x >> 16 & UINT128_C(0x0000ffff0000ffff0000ffff0000ffff));
      x = (x & UINT128_C(0x00000000ffffffff00000000ffffffff)) + (x >> 32 & UINT128_C(0x00000000ffffffff00000000ffffffff));
      x = (x & UINT128_C(0x0000000000000000ffffffffffffffff)) + (x >> 64 & UINT128_C(0x0000000000000000ffffffffffffffff));
      return x;
    }
  };
#endif
}


#line 5 "tools/uint128_t.hpp"


#line 5 "tools/prod_mod.hpp"

namespace tools {

  template <typename T1, typename T2, typename T3>
  constexpr T3 prod_mod(const T1 x, const T2 y, const T3 m) {
    using u128 = tools::uint128_t;
    u128 prod_mod = u128(x >= 0 ? x : -x) * u128(y >= 0 ? y : -y) % u128(m);
    if ((x >= 0) ^ (y >= 0)) prod_mod = u128(m) - prod_mod;
    return prod_mod;
  }
}


#line 1 "tools/pow_mod.hpp"



#line 1 "tools/mod.hpp"



#line 7 "tools/mod.hpp"

namespace tools {
  template <tools::non_bool_integral M, tools::non_bool_integral N>
  constexpr std::common_type_t<M, N> mod(const M a, const N b) noexcept {
    assert(b != 0);

    using UM = tools::make_unsigned_t<M>;
    using UN = tools::make_unsigned_t<N>;
    const UM ua = a >= 0 ? a : static_cast<UM>(-(a + 1)) + 1;
    const UN ub = b >= 0 ? b : static_cast<UN>(-(b + 1)) + 1;
    auto r = ua % ub;
    if (a < 0 && r > 0) {
      r = ub - r;
    }
    return r;
  }
}


#line 6 "tools/pow_mod.hpp"

namespace tools {

  template <typename T1, typename T2, typename T3>
  constexpr T3 pow_mod(const T1 x, T2 n, const T3 m) {
    if (m == 1) return 0;
    T3 r = 1;
    T3 y = tools::mod(x, m);
    while (n > 0) {
      if ((n & 1) > 0) {
        r = tools::prod_mod(r, y, m);
      }
      y = tools::prod_mod(y, y, m);
      n /= 2;
    }
    return r;
  }
}


#line 7 "tools/is_prime.hpp"

namespace tools {

  constexpr bool is_prime(const unsigned long long n) {
    constexpr std::array<unsigned long long, 7> bases = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};

    if (n <= 1) return false;
    if (n == 2) return true;
    if (n % 2 == 0) return false;

    auto d = n - 1;
    for (; d % 2 == 0; d /= 2);

    for (const auto a : bases) {
      if (a % n == 0) return true;

      auto power = d;
      auto target = tools::pow_mod(a, power, n);

      bool is_composite = true;
      if (target == 1) is_composite = false;
      for (; is_composite && power != n - 1; power *= 2, target = tools::prod_mod(target, target, n)) {
        if (target == n - 1) is_composite = false;
      }

      if (is_composite) {
        return false;
      }
    }

    return true;
  }
}


#line 1 "tools/monoids.hpp"



#line 14 "tools/monoids.hpp"

namespace tools {
  namespace monoids {
    template <typename M>
    struct bit_and {
      using T = M;
      static T op(const T& x, const T& y) {
        return x & y;
      }
      static T e() {
        return std::numeric_limits<T>::max();
      }
    };

    template <typename M>
    struct bit_or {
      using T = M;
      static T op(const T& x, const T& y) {
        return x | y;
      }
      static T e() {
        return T(0);
      }
    };

    template <typename M>
    requires requires (M x, M y) {
      {tools::gcd(x, y)} -> std::convertible_to<M>;
    }
    struct gcd {
      using T = M;
      static T op(const T& x, const T& y) {
        return tools::gcd(x, y);
      }
      static T e() {
        return T(0);
      }
    };

    template <typename M, M ...dummy>
    struct max;

    template <tools::arithmetic M>
    struct max<M> {
      using T = M;
      static T op(const T& x, const T& y) {
        return std::max(x, y);
      }
      static T e() {
        if constexpr (tools::integral<M>) {
          return std::numeric_limits<M>::min();
        } else {
          return -std::numeric_limits<M>::infinity();
        }
      }
    };

    template <std::totally_ordered M, M E>
    struct max<M, E> {
      using T = M;
      static T op(const T& x, const T& y) {
        assert(E <= x);
        assert(E <= y);
        return std::max(x, y);
      }
      static T e() {
        return E;
      }
    };

    template <typename M, M ...dummy>
    struct min;

    template <tools::arithmetic M>
    struct min<M> {
      using T = M;
      static T op(const T& x, const T& y) {
        return std::min(x, y);
      }
      static T e() {
        if constexpr (tools::integral<M>) {
          return std::numeric_limits<M>::max();
        } else {
          return std::numeric_limits<M>::infinity();
        }
      }
    };

    template <std::totally_ordered M, M E>
    struct min<M, E> {
      using T = M;
      static T op(const T& x, const T& y) {
        assert(x <= E);
        assert(y <= E);
        return std::min(x, y);
      }
      static T e() {
        return E;
      }
    };

    template <typename M>
    struct multiplies {
      using T = M;
      static T op(const T& x, const T& y) {
        return x * y;
      }
      static T e() {
        return T(1);
      }
    };

    template <>
    struct multiplies<bool> {
      using T = bool;
      static T op(const bool x, const bool y) {
        return x && y;
      }
      static T e() {
        return true;
      }
    };

    template <typename M, M E>
    struct update {
      using T = M;
      static T op(const T& x, const T& y) {
        return x == E ? y : x;
      }
      static T e() {
        return E;
      }
    };
  }
}


#line 1 "tools/prime_static_modint.hpp"



#line 6 "tools/prime_static_modint.hpp"

namespace tools {
  template <typename T>
  concept prime_static_modint = atcoder::internal::is_static_modint<T>::value && tools::is_prime(T::mod());
}


#line 13 "tools/multiplicative_structure.hpp"

namespace tools {
  template <typename T>
  using multiplicative_structure = std::conditional_t<
    tools::complex<T> || std::floating_point<T> || tools::prime_static_modint<T> || atcoder::internal::is_dynamic_modint<T>::value,
      tools::groups::multiplies<T>,
      std::conditional_t<
        tools::integral<T> || atcoder::internal::is_static_modint<T>::value,
          tools::monoids::multiplies<T>,
          std::conditional_t<
            requires(T a, T b) { { a / b } -> std::same_as<T>; },
              tools::groups::multiplies<T>,
              tools::monoids::multiplies<T>
          >
      >
  >;
}
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