Cache for $n^{-1}, n!, n!^{-1} \pmod{P}$ (tools/fact_mod_cache.hpp)

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• Last update: 2025-02-10 20:05:30+09:00
• Include: #include "tools/fact_mod_cache.hpp"

It caches $n^{-1}, n!, n!^{-1} \pmod{P}$ and returns them in $O(1)$ amortized time. It also returns ${}_n \mathrm{C}_r, {}_n \mathrm{P}_r, {}_n \mathrm{H}_r \pmod{P}$ in logarithmic amortized time.

License

• CC0

Author

• anqooqie

Constructor

(1) fact_mod_cache<M> cache();
(2) fact_mod_cache<M> cache(long long N);

• (1)
  • It creates an empty cache to store $n^{-1}, n!, n!^{-1} \pmod{P}$ where $P$ is M::mod().
• (2)
  • It precomputes $n^{-1}, n!, n!^{-1} \pmod{P}$ for any $n$ such that $0 \leq n \leq N$, where $P$ is M::mod().

Constraints

• <M> is atcoder::static_modint or atcoder::dynamic_modint
• M::mod() is a prime

Time Complexity

• (1)
  • $O(1)$
• (2)
  • $O(\min(N, P))$

inv

M cache.inv(long long n);

It returns $n^{-1} \pmod{P}$.

Constraints

• $n \not\equiv 0 \pmod{P}$

Time Complexity

• $O(\min(|n|, P))$ worst
• $O(1)$ amortized

fact

M cache.fact(long long n);

It returns $n! \pmod{P}$.

Constraints

• $n \geq 0$

Time Complexity

• $O(\min(n, P))$ worst
• $O(1)$ amortized

fact_inv

M cache.fact_inv(long long n);

It returns $n!^{-1} \pmod{P}$.

Constraints

• $0 < n < P$

Time Complexity

• $O(n)$ worst
• $O(1)$ amortized

binomial

M cache.binomial(long long n, long long r);

It returns $\dbinom{n}{r} \pmod{P}$. Note that $\dbinom{-n}{r} = (-1)^r \dbinom{n + r - 1}{r}$.

Constraints

• None

Time Complexity

• ($n \geq 0$):
  • $O(\min(n, P) + \log_P(n))$ worst
  • $O(\log_P(n))$ amortized
• ($n < 0$):
  • $O(\min(|n| + r, P) + \log_P(|n| + r))$ worst
  • $O(\log_P(|n| + r))$ amortized

combination

M cache.combination(long long n, long long r);

It returns

\[\begin{align*} \left\{\begin{array}{ll} {}_n \mathrm{C}_r \pmod{P} & \text{(if $0 \leq r \leq n$)}\\ 0 \pmod{P} & \text{(otherwise)} \end{array}\right.& \end{align*}\]

Constraints

• None

Time Complexity

• $O(\min(n, P) + \log_P(n))$ worst
• $O(\log_P(n))$ amortized

permutation

M cache.permutation(long long n, long long r);

It returns

\[\begin{align*} \left\{\begin{array}{ll} {}_n \mathrm{P}_r \pmod{P} & \text{(if $0 \leq r \leq n$)}\\ 0 \pmod{P} & \text{(otherwise)} \end{array}\right.& \end{align*}\]

Constraints

• None

Time Complexity

• $O(\min(n, P) + \log_P(n))$ worst
• $O(\log_P(n))$ amortized

combination_with_repetition

M cache.combination_with_repetition(long long n, long long r);

It returns

\[\begin{align*} \left\{\begin{array}{ll} {}_n \mathrm{H}_r \pmod{P} & \text{(if $n \geq 0$ and $r \geq 0$)}\\ 0 \pmod{P} & \text{(otherwise)} \end{array}\right.& \end{align*}\]

Constraints

• None

Time Complexity

• $O(\min(n + r, P) + \log_P(n + r))$ worst
• $O(\log_P(n + r))$ amortized

Depends on

• std::abs(x) extended for my library (tools/abs.hpp)
• $2^{\left\lceil \log_2(x) \right\rceil}$ (tools/bit_ceil.hpp)
• $2^{\left\lfloor \log_2(x) \right\rfloor}$ (tools/bit_floor.hpp)
• Bit width required to represent a given integer (tools/bit_width.hpp)
• Number of trailing zeros (tools/countr_zero.hpp)
• tools/detail/int128_t.hpp
• Combine hash values (tools/hash_combine.hpp)
• std::is_integral extended for tools::int128_t and tools::uint128_t (tools/is_integral.hpp)
• Miller-Rabin primality test (tools/is_prime.hpp)
• std::is_signed extended for tools::int128_t and tools::uint128_t (tools/is_signed.hpp)
• std::is_unsigned extended for tools::int128_t and tools::uint128_t (tools/is_unsigned.hpp)
• std::make_signed extended for tools::int128_t and tools::uint128_t (tools/make_signed.hpp)
• std::make_unsigned extended for tools::int128_t and tools::uint128_t (tools/make_unsigned.hpp)
• Minimum non-negative reminder (tools/mod.hpp)
• The number of nanoseconds that have elapsed since epoch (tools/now.hpp)
• $x^y \pmod{M}$ (tools/pow_mod.hpp)
• $x \cdot y \pmod{M}$ (tools/prod_mod.hpp)
• 128-bit unsigned integer (tools/uint128_t.hpp)

Required by

• Bell numbers (tools/bell.hpp)
• Berlekamp-Massey algorithm (tools/berlekamp_massey.hpp)
• Bernoulli numbers $B_k \pmod{P}$ for $0 \leq k \leq n$ (tools/bernoulli.hpp)
• Bostan-Mori algorithm (tools/bostan_mori.hpp)
• Precompute $n! \pmod{P}$ for $0 \leq n < P \approx 10^9$ (tools/large_fact_mod_cache.hpp)
• Polynomial (tools/polynomial.hpp)
• Polynomial interpolation (tools/polynomial_interpolation.hpp)
• Shift of sampling points of polynomial (tools/sample_point_shift.hpp)
• Power of a sparse FPS (tools/sparse_fps_pow.hpp)
• Signed Stirling numbers of the first kind (tools/stirling_1st.hpp)
• Stirling numbers of the second kind (tools/stirling_2nd.hpp)
• Twelvefold way (tools/twelvefold_way.hpp)

Verified with

• tests/bell/all.test.cpp
• tests/bell/consistent.test.cpp
• tests/bell/diagonal.test.cpp
• tests/bell/fixed_n.test.cpp
• tests/berlekamp_massey.test.cpp
• tests/bernoulli.test.cpp
• tests/bostan_mori.test.cpp
• tests/fact_mod_cache.test.cpp
• tests/large_fact_mod_cache/binomial.test.cpp
• tests/large_fact_mod_cache/fact.test.cpp
• tests/polynomial/multidimensional.test.cpp
• tests/polynomial/multipoint_evaluation.test.cpp
• tests/polynomial/multipoint_evaluation_other_mods.test.cpp
• tests/polynomial/naive_division.test.cpp
• tests/polynomial/ntt_division.test.cpp
• tests/polynomial/taylor_shift.test.cpp
• tests/polynomial_interpolation.test.cpp
• tests/polynomial_product.test.cpp
• tests/sample_point_shift.test.cpp
• tests/sparse_fps_pow/fraction.test.cpp
• tests/sparse_fps_pow/regular.test.cpp
• tests/stirling_1st/consistent.test.cpp
• tests/stirling_1st/fixed_k.test.cpp
• tests/stirling_1st/fixed_n.test.cpp
• tests/stirling_2nd/all.test.cpp
• tests/stirling_2nd/consistent.test.cpp
• tests/stirling_2nd/fixed_k.test.cpp
• tests/stirling_2nd/fixed_n.test.cpp
• tests/twelvefold_way/labeled_ball_labeled_box_at_least_1.test.cpp
• tests/twelvefold_way/labeled_ball_labeled_box_at_most_1.test.cpp
• tests/twelvefold_way/labeled_ball_labeled_box_unrestricted.test.cpp
• tests/twelvefold_way/labeled_ball_unlabeled_box_at_least_1.test.cpp
• tests/twelvefold_way/labeled_ball_unlabeled_box_at_most_1.test.cpp
• tests/twelvefold_way/labeled_ball_unlabeled_box_unrestricted.test.cpp
• tests/twelvefold_way/unlabeled_ball_labeled_box_at_least_1.test.cpp
• tests/twelvefold_way/unlabeled_ball_labeled_box_at_most_1.test.cpp
• tests/twelvefold_way/unlabeled_ball_labeled_box_unrestricted.test.cpp
• tests/twelvefold_way/unlabeled_ball_unlabeled_box_at_least_1.test.cpp
• tests/twelvefold_way/unlabeled_ball_unlabeled_box_at_most_1.test.cpp
• tests/twelvefold_way/unlabeled_ball_unlabeled_box_unrestricted.test.cpp

Code

#ifndef TOOLS_FACT_MOD_CACHE_HPP
#define TOOLS_FACT_MOD_CACHE_HPP

#include <vector>
#include <cassert>
#include <iterator>
#include <algorithm>
#include <cmath>
#include "tools/is_prime.hpp"

namespace tools {

  template <class M>
  class fact_mod_cache {
    ::std::vector<M> m_inv;
    ::std::vector<M> m_fact;
    ::std::vector<M> m_fact_inv;

  public:
    fact_mod_cache() : m_inv({M::raw(0), M::raw(1)}), m_fact({M::raw(1), M::raw(1)}), m_fact_inv({M::raw(1), M::raw(1)}) {
      assert(::tools::is_prime(M::mod()));
    }
    explicit fact_mod_cache(const long long max) : fact_mod_cache() {
      this->fact(::std::min<long long>(max, M::mod() - 1));
      this->fact_inv(::std::min<long long>(max, M::mod() - 1));
    }

    M inv(const long long n) {
      assert(n % M::mod() != 0);
      const long long size = ::std::ssize(this->m_inv);
      this->m_inv.resize(::std::clamp<long long>(::std::abs(n) + 1, size, M::mod()));
      for (long long i = size; i < ::std::ssize(this->m_inv); ++i) {
        this->m_inv[i] = -this->m_inv[M::mod() % i] * M::raw(M::mod() / i);
      }
      M result = this->m_inv[::std::abs(n) % M::mod()];
      if (n < 0) result = -result;
      return result;
    }
    M fact(const long long n) {
      assert(n >= 0);
      const long long size = ::std::ssize(this->m_fact);
      this->m_fact.resize(::std::clamp<long long>(n + 1, size, M::mod()));
      for (long long i = size; i < ::std::ssize(this->m_fact); ++i) {
        this->m_fact[i] = this->m_fact[i - 1] * M::raw(i);
      }
      return n < M::mod() ? this->m_fact[n] : M::raw(0);
    }
    M fact_inv(const long long n) {
      assert(0 <= n && n < M::mod());
      const long long size = ::std::ssize(this->m_fact_inv);
      this->m_fact_inv.resize(::std::max<long long>(size, n + 1));
      this->inv(this->m_fact_inv.size() - 1);
      for (long long i = size; i < ::std::ssize(this->m_fact_inv); ++i) {
        this->m_fact_inv[i] = this->m_fact_inv[i - 1] * this->m_inv[i];
      }
      return this->m_fact_inv[n];
    }

    M binomial(long long n, long long r) {
      if (r < 0) return M::raw(0);
      if (0 <= n && n < r) return M::raw(0);
      if (n < 0) return M(1 - ((r & 1) << 1)) * this->binomial(-n + r - 1, r);

      this->fact(::std::min<long long>(n, M::mod() - 1));
      this->fact_inv(::std::min<long long>(n, M::mod() - 1));
      const auto c = [&](const long long nn, const long long rr) {
        return 0 <= rr && rr <= nn ? this->m_fact[nn] * this->m_fact_inv[nn - rr] * this->m_fact_inv[rr] : M::raw(0);
      };

      M result(1);
      while (n > 0 || r > 0) {
        result *= c(n % M::mod(), r % M::mod());
        n /= M::mod();
        r /= M::mod();
      }

      return result;
    }
    M combination(const long long n, const long long r) {
      if (!(0 <= r && r <= n)) return M::raw(0);
      return this->binomial(n, r);
    }
    M permutation(const long long n, const long long r) {
      if (!(0 <= r && r <= n)) return M::raw(0);
      return this->binomial(n, r) * this->fact(r);
    }
    M combination_with_repetition(const long long n, const long long r) {
      if (n < 0 || r < 0) return M::raw(0);
      return this->binomial(n + r - 1, r);
    }
  };
}

#endif

#line 1 "tools/fact_mod_cache.hpp"



#include <vector>
#include <cassert>
#include <iterator>
#include <algorithm>
#include <cmath>
#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.hpp"



#line 6 "tools/detail/int128_t.hpp"
#include <cstddef>
#include <cstdint>
#include <functional>
#include <iostream>
#include <limits>
#include <string>
#include <string_view>
#line 1 "tools/abs.hpp"



namespace tools {
  constexpr float abs(const float x) {
    return x < 0 ? -x : x;
  }
  constexpr double abs(const double x) {
    return x < 0 ? -x : x;
  }
  constexpr long double abs(const long double x) {
    return x < 0 ? -x : x;
  }
  constexpr int abs(const int x) {
    return x < 0 ? -x : x;
  }
  constexpr long abs(const long x) {
    return x < 0 ? -x : x;
  }
  constexpr long long abs(const long long x) {
    return x < 0 ? -x : x;
  }
  constexpr unsigned int abs(const unsigned int x) {
    return x;
  }
  constexpr unsigned long abs(const unsigned long x) {
    return x;
  }
  constexpr unsigned long long abs(const unsigned long long x) {
    return x;
  }
}


#line 1 "tools/bit_ceil.hpp"



#include <bit>
#line 6 "tools/bit_ceil.hpp"
#include <type_traits>
#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 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/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 10 "tools/bit_ceil.hpp"

namespace tools {
  template <typename T>
  constexpr T bit_ceil(T) noexcept;

  template <typename T>
  constexpr T bit_ceil(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_ceil<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_ceil(x);
    }
  }
}


#line 1 "tools/bit_floor.hpp"



#line 10 "tools/bit_floor.hpp"

namespace tools {
  template <typename T>
  constexpr T bit_floor(T) noexcept;

  template <typename T>
  constexpr T bit_floor(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_floor<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_floor(x);
    }
  }
}


#line 1 "tools/bit_width.hpp"



#line 10 "tools/bit_width.hpp"

namespace tools {
  template <typename T>
  constexpr int bit_width(T) noexcept;

  template <typename T>
  constexpr int bit_width(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::tools::bit_width<::tools::make_unsigned_t<T>>(x);
    } else {
      return ::std::bit_width(x);
    }
  }
}


#line 1 "tools/countr_zero.hpp"



#line 12 "tools/countr_zero.hpp"

namespace tools {
  template <typename T>
  constexpr int countr_zero(const T x) noexcept {
    static_assert(::tools::is_integral_v<T> && !::std::is_same_v<::std::remove_cv_t<T>, bool>);
    if constexpr (::tools::is_signed_v<T>) {
      assert(x >= 0);
      return ::std::min(::tools::countr_zero<::tools::make_unsigned_t<T>>(x), ::std::numeric_limits<T>::digits);
    } else {
      return ::std::countr_zero(x);
    }
  }
}


#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/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 25 "tools/detail/int128_t.hpp"

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

  namespace detail {
    namespace 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;
      }
    }
  }

  constexpr ::tools::uint128_t abs(const ::tools::uint128_t& x) noexcept {
    return x;
  }
  constexpr ::tools::int128_t abs(const ::tools::int128_t& x) {
    return x >= 0 ? x : -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 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;
  };

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
  template <>
  constexpr ::tools::uint128_t bit_ceil<::tools::uint128_t>(::tools::uint128_t x) 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 <>
  constexpr ::tools::uint128_t bit_floor<::tools::uint128_t>(::tools::uint128_t x) 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 <>
  constexpr int bit_width<::tools::uint128_t>(::tools::uint128_t x) 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;
  }

  namespace detail {
    namespace countr_zero {
      template <::std::size_t N>
      struct ntz_traits;

      template <>
      struct ntz_traits<128> {
        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
        };
      };

      template <typename T>
      constexpr int impl(const T x) noexcept {
        using tr = ::tools::detail::countr_zero::ntz_traits<::std::numeric_limits<T>::digits>;
        using type = typename tr::type;
        return tr::ntz_table[static_cast<type>(tr::magic * static_cast<type>(x & -x)) >> tr::shift];
      }
    }
  }

  template <>
  constexpr int countr_zero<::tools::uint128_t>(const ::tools::uint128_t x) noexcept {
    return ::tools::detail::countr_zero::impl(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 <typename M, typename N> requires (
    ::tools::is_integral_v<M> && !::std::is_same_v<::std::remove_cv_t<M>, bool> &&
    ::tools::is_integral_v<N> && !::std::is_same_v<::std::remove_cv_t<N>, bool>)
  constexpr ::std::common_type_t<M, N> mod(const M a, const N b) noexcept {
    assert(b != 0);

    using UM = ::std::make_unsigned_t<M>;
    using UN = ::std::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 10 "tools/fact_mod_cache.hpp"

namespace tools {

  template <class M>
  class fact_mod_cache {
    ::std::vector<M> m_inv;
    ::std::vector<M> m_fact;
    ::std::vector<M> m_fact_inv;

  public:
    fact_mod_cache() : m_inv({M::raw(0), M::raw(1)}), m_fact({M::raw(1), M::raw(1)}), m_fact_inv({M::raw(1), M::raw(1)}) {
      assert(::tools::is_prime(M::mod()));
    }
    explicit fact_mod_cache(const long long max) : fact_mod_cache() {
      this->fact(::std::min<long long>(max, M::mod() - 1));
      this->fact_inv(::std::min<long long>(max, M::mod() - 1));
    }

    M inv(const long long n) {
      assert(n % M::mod() != 0);
      const long long size = ::std::ssize(this->m_inv);
      this->m_inv.resize(::std::clamp<long long>(::std::abs(n) + 1, size, M::mod()));
      for (long long i = size; i < ::std::ssize(this->m_inv); ++i) {
        this->m_inv[i] = -this->m_inv[M::mod() % i] * M::raw(M::mod() / i);
      }
      M result = this->m_inv[::std::abs(n) % M::mod()];
      if (n < 0) result = -result;
      return result;
    }
    M fact(const long long n) {
      assert(n >= 0);
      const long long size = ::std::ssize(this->m_fact);
      this->m_fact.resize(::std::clamp<long long>(n + 1, size, M::mod()));
      for (long long i = size; i < ::std::ssize(this->m_fact); ++i) {
        this->m_fact[i] = this->m_fact[i - 1] * M::raw(i);
      }
      return n < M::mod() ? this->m_fact[n] : M::raw(0);
    }
    M fact_inv(const long long n) {
      assert(0 <= n && n < M::mod());
      const long long size = ::std::ssize(this->m_fact_inv);
      this->m_fact_inv.resize(::std::max<long long>(size, n + 1));
      this->inv(this->m_fact_inv.size() - 1);
      for (long long i = size; i < ::std::ssize(this->m_fact_inv); ++i) {
        this->m_fact_inv[i] = this->m_fact_inv[i - 1] * this->m_inv[i];
      }
      return this->m_fact_inv[n];
    }

    M binomial(long long n, long long r) {
      if (r < 0) return M::raw(0);
      if (0 <= n && n < r) return M::raw(0);
      if (n < 0) return M(1 - ((r & 1) << 1)) * this->binomial(-n + r - 1, r);

      this->fact(::std::min<long long>(n, M::mod() - 1));
      this->fact_inv(::std::min<long long>(n, M::mod() - 1));
      const auto c = [&](const long long nn, const long long rr) {
        return 0 <= rr && rr <= nn ? this->m_fact[nn] * this->m_fact_inv[nn - rr] * this->m_fact_inv[rr] : M::raw(0);
      };

      M result(1);
      while (n > 0 || r > 0) {
        result *= c(n % M::mod(), r % M::mod());
        n /= M::mod();
        r /= M::mod();
      }

      return result;
    }
    M combination(const long long n, const long long r) {
      if (!(0 <= r && r <= n)) return M::raw(0);
      return this->binomial(n, r);
    }
    M permutation(const long long n, const long long r) {
      if (!(0 <= r && r <= n)) return M::raw(0);
      return this->binomial(n, r) * this->fact(r);
    }
    M combination_with_repetition(const long long n, const long long r) {
      if (n < 0 || r < 0) return M::raw(0);
      return this->binomial(n + r - 1, r);
    }
  };
}

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