#933413 (C++17) No.502 階乗を計算するだけ

• 問題: No.502 階乗を計算するだけ
• ユーザー: miscalc
• 提出日時: 2023-12-02 04:01:20
• 言語: C++17 (gcc 13.3.0 + boost 1.87.0)
• 結果: AC
• 実行時間: 60 ms / 1,000 ms
• コード長: 50,107 bytes
• コンパイル時間: 3,839 ms
• コンパイル使用メモリ: 257,212 KB
• 最終ジャッジ日時: 2025-02-18 03:19:46
• ジャッジサーバーID （参考情報）: judge5 / judge2

ファイルパターン	結果
other	AC * 52

ソースコード

#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using ld = long double;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using tlll = tuple<ll, ll, ll>;
constexpr ll INF = 1LL << 60;
template<class T> bool chmin(T& a, T b) {if (a > b) {a = b; return true;} return false;}
template<class T> bool chmax(T& a, T b) {if (a < b) {a = b; return true;} return false;}
ll safemod(ll A, ll M) {ll res = A % M; if (res < 0) res += M; return res;}
ll divfloor(ll A, ll B) {if (B < 0) A = -A, B = -B; return (A - safemod(A, B)) / B;}
ll divceil(ll A, ll B) {if (B < 0) A = -A, B = -B; return divfloor(A + B - 1, B);}
ll pow_ll(ll A, ll B) {if (A == 0 || A == 1) {return A;} if (A == -1) {return B & 1 ? -1 : 1;} ll res = 1; for (int i = 0; i < B; i++) {res *= A;} return res;}
ll mul_limited(ll A, ll B, ll M = INF) { return B == 0 ? 0 : A > M / B ? M : A * B; }
ll pow_limited(ll A, ll B, ll M = INF) { if (A == 0 || A == 1) {return A;} ll res = 1; for (int i = 0; i < B; i++) {if (res > M / A) return M; res *= A;} return res;}
template<class T> void unique(vector<T> &V) {V.erase(unique(V.begin(), V.end()), V.end());}
template<class T> void sortunique(vector<T> &V) {sort(V.begin(), V.end()); V.erase(unique(V.begin(), V.end()), V.end());}
#define FINALANS(A) do {cout << (A) << '\n'; exit(0);} while (false)
template<class T> void printvec(const vector<T> &V) {int _n = V.size(); for (int i = 0; i < _n; i++) cout << V[i] << (i == _n - 1 ? "" : " ");cout << '\n';}
template<class T> void printvect(const vector<T> &V) {for (auto v : V) cout << v << '\n';}
template<class T> void printvec2(const vector<vector<T>> &V) {for (auto &v : V) printvec(v);}
//*
#include <atcoder/modint>
#include <atcoder/math>
#include <atcoder/convolution>
#include <atcoder/internal_math>
using namespace atcoder;
//*/

// http://drken1215.hatenablog.com/entry/2018/06/08/210000
template <class T>
class binom
{
public:
  vector<T> fac, finv, inv;

  binom(int M)
  {
    fac.resize(M + 1);
    finv.resize(M + 1);
    inv.resize(M + 1);

    //*
    fac[0] = T(1);
    for (int i = 1; i <= M; i++)
      fac[i] = fac[i - 1] * T::raw(i);
    finv[M] = fac[M].inv();
    for (int i = M - 1; i >= 0; i--)
      finv[i] = finv[i + 1] * T::raw(i + 1);
    for (int i = 1; i <= M; i++)
      inv[i] = fac[i - 1] * finv[i];
    //*/

    /*
    fac[0] = T(1), finv[0] = T(1);
    fac[1] = T(1), finv[1] = T(1), inv[1] = T(1);
    for (int i = 2; i <= M; i++)
    {
      fac[i] = fac[i - 1] * i;
      inv[i] = -inv[T::mod() % i] * (T::mod() / i);
      finv[i] = finv[i - 1] * inv[i];
    }
    //*/
  }

  T P(int N, int K)
  {
    if (N < K) return 0;
    if (N < 0 || K < 0) return 0;
    return fac[N] * finv[N - K];
  }

  T C(int N, int K)
  {
    if (N < K) return 0;
    if (N < 0 || K < 0) return 0;
    return fac[N] * finv[K] * finv[N - K];
  }

  T H(int N, int K)
  {
    if (N == 0 && K == 0)
      return 1;
    return C(N + K - 1, K);
  }
};

// http://drken1215.hatenablog.com/entry/2018/06/08/210000
template <class T>
class binom_mut
{
private:
  vector<T> fac, finv, inv;
  void calc(int n)
  {
    int i = fac.size();
    if (n < i)
      return;
    fac.resize(n + 1), finv.resize(n + 1), inv.resize(n + 1);
    for (; i <= n; i++)
    {
      fac[i] = fac[i - 1] * i;
      inv[i] = -inv[T::mod() % i] * (T::mod() / i);
      finv[i] = finv[i - 1] * inv[i];
    }
  }
public:
  binom_mut() { fac = {1, 1}, finv = {1, 1}, inv = {0, 1}; }

  T get_fac(int n) { assert(n >= 0); calc(n); return fac[n]; }
  T get_finv(int n) { assert(n >= 0); calc(n); return finv[n]; }
  T get_inv(int n) { assert(n > 0); calc(n); return inv[n]; }

  T P(int N, int K)
  {
    if (N < K) return 0;
    if (N < 0 || K < 0) return 0;
    calc(N);
    return fac[N] * finv[N - K];
  }
  T C(int N, int K)
  {
    if (N < K) return 0;
    if (N < 0 || K < 0) return 0;
    calc(N);
    return fac[N] * finv[K] * finv[N - K];
  }
  T H(int N, int K)
  {
    if (N == 0 && K == 0)
      return 1;
    return C(N + K - 1, K);
  }
};

// https://qiita.com/taiyaki8926/items/f62f534d43ff006129f7
ll sqrt_mod(ll n, int p) // p は素数
{
  n %= p;
  if (n == 0)
    return 0;
  if (p == 2)
    return n;
  if (pow_mod(n, (p - 1) / 2, p) == p - 1) // 平方非剰余
    return -1;
  if (p % 4 == 3)
    return pow_mod(n, (p + 1) / 4, p);

  internal::barrett ba(p);
  int q = p - 1, s = 0;
  while (q % 2 == 0)
    q /= 2, s++;
  int z = 2;
  while (pow_mod(z, (p - 1) / 2, p) != p - 1)
    z++;
  int m = s;
  ll c = pow_mod(z, q, p);
  ll t = pow_mod(n, q, p);
  ll r = pow_mod(n, (q + 1) / 2, p);
  while (t != 1)
  {
    int m2 = 1;
    for (ll tmp = ba.mul(t, t); tmp != 1; tmp = ba.mul(tmp, tmp), m2++);
    ll b = pow_mod(c, 1 << (m - m2 - 1), p);
    m = m2, c = ba.mul(b, b), t = ba.mul(t, c), r = ba.mul(r, b);
  }
  return r;
}

template<const int MOD = 1000000007, class T>
vector<T> convolution_anymod(const vector<T> &A, const vector<T> &B)
{
  int N = A.size(), M = B.size();
  if (min(N, M) <= 300)
  {
    using mint = static_modint<MOD>;
    vector<mint> A2(N), B2(M);
    for (int i = 0; i < N; i++)
      A2[i] = A[i];
    for (int j = 0; j < M; j++)
      B2[j] = B[j];
    vector<mint> C2(N + M - 1, 0);
    for (int i = 0; i < N; i++)
      for (int j = 0; j < M; j++)
        C2[i + j] += A2[i] * B2[j];
    vector<T> C(N + M - 1);
    for (int i = 0; i < N + M - 1; i++)
      C[i] = C2[i].val();
    return C;
  }

  constexpr ll MOD1 = 167772161, MOD2 = 469762049, MOD3 = 1224736769;
  using mint2 = static_modint<MOD2>;
  using mint3 = static_modint<MOD3>;
  using mint4 = static_modint<MOD>;
  constexpr int i1_2 = internal::inv_gcd(MOD1, MOD2).second;
  constexpr int i12_3 = internal::inv_gcd(MOD1 * MOD2, MOD3).second;
  constexpr int m12_4 = MOD1 * MOD2 % MOD;

  auto C1 = convolution<MOD1>(A, B);
  auto C2 = convolution<MOD2>(A, B);
  auto C3 = convolution<MOD3>(A, B);

  vector<T> C(N + M - 1);
  for (ll i = 0; i < N + M - 1; i++)
  {
    int c1 = C1[i], c2 = C2[i], c3 = C3[i];
    int t1 = (mint2(c2 - c1) * mint2::raw(i1_2)).val();
    mint3 x2_m3 = mint3::raw(c1) + mint3::raw(t1) * mint3::raw(MOD1);
    mint4 x2_m = mint4::raw(c1) + mint4::raw(t1) * mint4::raw(MOD1);
    int t2 = ((mint3::raw(c3) - x2_m3) * mint3::raw(i12_3)).val();
    C[i] = (x2_m + mint4::raw(t2) * mint4::raw(m12_4)).val();
  }
  return C;
}
template<class T>
vector<T> convolution_anymod(const vector<T> &A, const vector<T> &B, const int MOD)
{
  int N = A.size(), M = B.size();
  if (min(N, M) <= 300)
  {
    using mint = dynamic_modint<100>;
    mint::set_mod(MOD);
    vector<mint> A2(N), B2(M);
    for (int i = 0; i < N; i++)
      A2[i] = A[i];
    for (int j = 0; j < M; j++)
      B2[j] = B[j];
    vector<mint> C2(N + M - 1, 0);
    for (int i = 0; i < N; i++)
      for (int j = 0; j < M; j++)
        C2[i + j] += A2[i] * B2[j];
    vector<T> C(N + M - 1);
    for (int i = 0; i < N + M - 1; i++)
      C[i] = C2[i].val();
    return C;
  }

  constexpr ll MOD1 = 167772161, MOD2 = 469762049, MOD3 = 1224736769;
  using mint2 = static_modint<MOD2>;
  using mint3 = static_modint<MOD3>;
  using mint4 = dynamic_modint<100>;
  mint4::set_mod(MOD);
  constexpr int i1_2 = internal::inv_gcd(MOD1, MOD2).second;
  constexpr int i12_3 = internal::inv_gcd(MOD1 * MOD2, MOD3).second;

  auto C1 = convolution<MOD1>(A, B);
  auto C2 = convolution<MOD2>(A, B);
  auto C3 = convolution<MOD3>(A, B);

  vector<T> C(N + M - 1);
  for (ll i = 0; i < N + M - 1; i++)
  {
    int c1 = C1[i], c2 = C2[i], c3 = C3[i];
    int t1 = (mint2(c2 - c1) * mint2::raw(i1_2)).val();
    mint3 x2_m3 = mint3::raw(c1) + mint3::raw(t1) * mint3::raw(MOD1);
    mint4 x2_m = mint4::raw(c1) + mint4::raw(t1) * mint4::raw(MOD1);
    int t2 = ((mint3::raw(c3) - x2_m3) * mint3::raw(i12_3)).val();
    C[i] = (x2_m + mint4::raw(t2) * mint4::raw(MOD1) * mint4::raw(MOD2)).val();
  }
  return C;
}
template<const int MOD>
vector<static_modint<MOD>> convolution_anymod(const vector<static_modint<MOD>> &A, const vector<static_modint<MOD>> &B)
{
  int N = A.size(), M = B.size();
  vector<int> A2(N), B2(M);
  for (int i = 0; i < N; i++)
    A2[i] = A[i].val();
  for (int i = 0; i < M; i++)
    B2[i] = B[i].val();
  vector<int> C2 = convolution_anymod<MOD>(A2, B2);
  vector<static_modint<MOD>> C(N + M - 1);
  for (int i = 0; i < N + M - 1; i++)
    C[i] = static_modint<MOD>::raw(C2[i]);
  return C;
}
template<const int id>
vector<dynamic_modint<id>> convolution_anymod(const vector<dynamic_modint<id>> &A, const vector<dynamic_modint<id>> &B)
{
  int N = A.size(), M = B.size();
  vector<int> A2(N), B2(M);
  for (int i = 0; i < N; i++)
    A2[i] = A[i].val();
  for (int i = 0; i < M; i++)
    B2[i] = B[i].val();
  vector<int> C2 = convolution_anymod(A2, B2, dynamic_modint<id>::mod());
  vector<dynamic_modint<id>> C(N + M - 1);
  for (int i = 0; i < N + M - 1; i++)
    C[i] = dynamic_modint<id>::raw(C2[i]);
  return C;
}

template<class T1>
struct LagrangeInterpolation
{
  int D;
  vector<T1> Y, fac, finv, prodl, prodr;
  
  template<class T2>
  LagrangeInterpolation(const vector<T2> &y)
  {
    D = (int)y.size() - 1;
    Y.resize(D + 1);
    for (int i = 0; i <= D; i++)
    {
      Y[i] = y[i];
    }

    fac.resize(D + 1), finv.resize(D + 1);
    fac[0] = 1;
    for (int i = 1; i <= D; i++)
      fac[i] = fac[i - 1] * i;
    finv[D] = fac[D].inv();
    for (int i = D - 1; i >= 0; i--)
      finv[i] = finv[i + 1] * (i + 1);

    prodl.resize(D + 2), prodr.resize(D + 2);
  }

  T1 eval(T1 x)
  {
    prodl[0] = 1;
    for (int i = 0; i <= D; i++)
    {
      prodl[i + 1] = prodl[i] * (x - i);
    }
    prodr[D + 1] = 1;
    for (int i = D; i >= 0; i--)
    {
      prodr[i] = prodr[i + 1] * (x - i);
    }

    T1 res = 0;
    for (int i = 0; i <= D; i++)
    {
      T1 tmp = Y[i] * prodl[i] * prodr[i + 1] * finv[i] * finv[D - i];
      if ((D - i) % 2 == 0)
        res += tmp;
      else
        res -= tmp;
    }
    return res;
  }
};

// https://opt-cp.com/fps-implementation/
// https://qiita.com/hotman78/items/f0e6d2265badd84d429a
// https://opt-cp.com/fps-fast-algorithms/
// https://maspypy.com/%E5%A4%9A%E9%A0%85%E5%BC%8F%E3%83%BB%E5%BD%A2%E5%BC%8F%E7%9A%84%E3%81%B9%E3%81%8D%E7%B4%9A%E6%95%B0-%E9%AB%98%E9%80%9F%E3%81%AB%E8%A8%88%E7%AE%97%E3%81%A7%E3%81%8D%E3%82%8B%E3%82%82%E3%81%AE
template<class T, bool is_ntt_friendly>
struct FormalPowerSeries : vector<T>
{
private:
  static vector<T> fac, finv, invmint;
  void calc(int n)
  {
    while ((int)fac.size() <= n)
    {
      int i = fac.size();
      fac.emplace_back(fac[i - 1] * i);
      invmint.emplace_back(-invmint[T::mod() % i] * (T::mod() / i));
      finv.emplace_back(finv[i - 1] * invmint[i]);
    }
  }

public:
  T get_fac(int n) { calc(n); return fac[n]; }
  T get_finv(int n) { calc(n); return finv[n]; }
  T get_invmint(int n) { calc(n); return invmint[n]; }

  using vector<T>::vector;
  using vector<T>::operator=;
  using F = FormalPowerSeries;
  using S = vector<pair<ll, T>>;

  FormalPowerSeries(const S &f, int n = -1)
  {
    if (n == -1)
      n = f.back().first + 1;
    (*this).assign(n, T(0));
    for (auto [d, a] : f)
      (*this)[d] += a;
  }

  F operator-() const
  {
    F res(*this);
    for (auto &a : res)
      a = -a;
    return res;
  }

  F operator*=(const T &k)
  {
    for (auto &a : *this)
      a *= k;
    return *this;
  }
  F operator*(const T &k) const { return F(*this) *= k; }
  friend F operator*(const T k, const F &f) { return f * k; }
  F operator/=(const T &k)
  {
    *this *= k.inv();
    return *this;
  }
  F operator/(const T &k) const { return F(*this) /= k; }

  F &operator+=(const F &g)
  {
    int n = (*this).size(), m = g.size();
    (*this).resize(max(n, m), T(0));
    for (int i = 0; i < m; i++)
      (*this)[i] += g[i];
    return *this;
  }
  F operator+(const F &g) const { return F(*this) += g; }
  F &operator-=(const F &g)
  {
    int n = (*this).size(), m = g.size();
    (*this).resize(max(n, m), T(0));
    for (int i = 0; i < m; i++)
      (*this)[i] -= g[i];
    return *this;
  }
  F operator-(const F &g) const { return F(*this) -= g; }

  F &operator<<=(const ll d)
  {
    int n = (*this).size();
    (*this).insert((*this).begin(), min(ll(n), d), T(0));
    (*this).resize(n);
    return *this;
  }
  F operator<<(const ll d) const { return F(*this) <<= d; }
  F &operator>>=(const ll d)
  {
    int n = (*this).size();
    (*this).erase((*this).begin(), (*this).begin() + min(ll(n), d));
    (*this).resize(n, T(0));
    return *this;
  }
  F operator>>(const ll d) const { return F(*this) >>= d; }

  F &operator*=(const S &g)
  {
    int n = (*this).size();
    auto [d, c] = g.front();
    if (d != 0)
      c = 0;
    for (int i = n - 1; i >= 0; i--)
    {
      (*this)[i] *= c;
      for (auto &[j, b] : g)
      {
        if (j == 0)
          continue;
        if (j > i)
          break;
        (*this)[i] += (*this)[i - j] * b;
      }
    }
    return *this;
  }
  F operator*(const S &g) const { return F(*this) *= g; }
  F &operator/=(const S &g)
  {
    int n = (*this).size();
    auto [d, c] = g.front();
    assert(d == 0 && c != T(0));
    T inv_c = c.inv();
    for (int i = 0; i < n; i++)
    {
      for (auto &[j, b] : g)
      {
        if (j == 0)
          continue;
        if (j > i)
          break;
        (*this)[i] -= (*this)[i - j] * b;
      }
      (*this)[i] *= inv_c;
    }
    return *this;
  }
  F operator/(const S &g) const { return F(*this) /= g; }

  // (1 + cx^d) を掛ける
  F multiply(const int d, const T c)
  {
    int n = (*this).size();
    if (c == T(1))
    {
      for (int i = n - 1 - d; i >= 0; i--)
        (*this)[i + d] += (*this)[i];
    }
    else if (c == T(-1))
    {
      for (int i = n - 1 - d; i >= 0; i--)
        (*this)[i + d] -= (*this)[i];
    }
    else
    {
      for (int i = n - 1 - d; i >= 0; i--)
        (*this)[i + d] += (*this)[i] * c;
    }
    return *this;
  }
  F multiplication(const int d, const T c) const { return multiply(F(*this)); }
  // (1 + cx^d) で割る
  F divide(const int d, const T c)
  {
    int n = (*this).size();
    if (c == T(1))
    {
      for (int i = 0; i < n - d; i++)
        (*this)[i + d] -= (*this)[i];
    }
    else if (c == T(-1))
    {
      for (int i = 0; i < n - d; i++)
        (*this)[i + d] += (*this)[i];
    }
    else
    {
      for (int i = 0; i < n - d; i++)
        (*this)[i + d] -= (*this)[i] * c;
    }
    return *this;
  }
  F division(const int d, const T c) const { return divide(F(*this)); }

  template<const int MOD>
  F convolution2(const vector<static_modint<MOD>> &A, const vector<static_modint<MOD>> &B, const int d = -1) const
  {
    F res;
    if (is_ntt_friendly)
      res = convolution(A, B);
    else
      res = convolution_anymod(A, B);
    if (d != -1 && (int)res.size() > d)
      res.resize(d);
    return res;
  }
  template<const int id>
  F convolution2(const vector<dynamic_modint<id>> &A, const vector<dynamic_modint<id>> &B, const int d = -1) const
  {
    F res;
    res = convolution_anymod(A, B);
    if (d != -1 && (int)res.size() > d)
      res.resize(d);
    return res;
  }

  F &operator*=(const F &g)
  {
    int n = (*this).size();
    if (n == 0)
      return *this;
    *this = convolution2(*this, g, n);
    return *this;
  }
  F operator*(const F &g) const { return F(*this) *= g; }

  template <const int MOD>
  void butterfly2(FormalPowerSeries<static_modint<MOD>, true> &A) const { internal::butterfly(A); }
  template <const int MOD>
  void butterfly2(FormalPowerSeries<static_modint<MOD>, false> &A) const { assert(false); }
  template <const int id>
  void butterfly2(FormalPowerSeries<dynamic_modint<id>, false> &A) const { assert(false); }
  template <const int MOD>
  void butterfly_inv2(FormalPowerSeries<static_modint<MOD>, true> &A) const { internal::butterfly_inv(A); }
  template <const int MOD>
  void butterfly_inv2(FormalPowerSeries<static_modint<MOD>, false> &A) const { assert(false); }
  template <const int id>
  void butterfly_inv2(FormalPowerSeries<dynamic_modint<id>, false> &A) const { assert(false); }

  // mod (x^n - 1) をとったものを返す
  F circular_mod(int n) const
  {
    F res(n, T(0));
    for (int i = 0; i < (int)(*this).size(); i++)
      res[i % n] += (*this)[i];
    return res;
  }

  F inv(int d = -1) const
  {
    int n = (*this).size();
    assert(!(*this).empty() && (*this).at(0) != T(0));
    if (d == -1)
      d = n;
    //assert(d > 0);
    F f, g2;
    F g{(*this).front().inv()};
    while ((int)g.size() < d)
    {
      if (is_ntt_friendly)
      {
        int m = g.size();
        f = F{(*this).begin(), (*this).begin() + min(n, 2 * m)};
        g2 = F(g);
        f.resize(2 * m, T(0)), butterfly2(f);
        g2.resize(2 * m, T(0)), butterfly2(g2);
        for (int i = 0; i < 2 * m; i++)
          f[i] *= g2[i];
        butterfly_inv2(f);
        f.erase(f.begin(), f.begin() + m);
        f.resize(2 * m, T(0)), butterfly2(f);
        for (int i = 0; i < 2 * m; i++)
          f[i] *= g2[i];
        butterfly_inv2(f);
        T iz = T(2 * m).inv();
        iz *= -iz;
        for (int i = 0; i < m; i++)
          f[i] *= iz;
        g.insert(g.end(), f.begin(), f.begin() + m);
      }
      else
      {
        g.resize(2 * g.size(), T(0));
        g *= F{T(2)} - g * (*this);
      }
    }
    return {g.begin(), g.begin() + d};
  }
  F &operator/=(const F &g)
  {
    *this *= g.inv((*this).size());
    return *this;
  }
  F operator/(const F &g) const { return F(*this) *= g.inv((*this).size()); }

  F differentiate()
  {
    *this >>= 1;
    for (int i = 0; i < int((*this).size()) - 1; i++)
      (*this)[i] *= i + 1;
    return *this;
  }
  F differential() const { return F(*this).differentiate(); }
  F integrate()
  {
    int n = (*this).size();
    vector<T> minv(n);
    minv[1] = T(1);
    *this <<= 1;
    for (int i = 2; i < n; i++)
    {
      minv[i] = -minv[T::mod() % i] * (T::mod() / i);
      (*this)[i] *= minv[i];
    }
    return *this;
  }
  F integral() const { return F(*this).integrate(); }

  F log() const
  {
    assert((*this).front() == T(1));
    return ((*this).differential() / (*this)).integral();
  }
  F exp() const // https://arxiv.org/pdf/1301.5804.pdf
  {
    int n = (*this).size();
    assert(n != 0 && (*this).front() == T(0));

    //*
    if (is_ntt_friendly)
    {
      F f{T(1)}, g{T(1)};
      F dh = (*this).differential();
      F f2, g2, f3, q, s, h, u;
      g2 = {T(0)};
      while ((int)f.size() < n)
      {
        int m = f.size();
        T im = T(m).inv(), i2m = T(2 * m).inv();
        f2 = F(f);
        f2.resize(2 * m), butterfly2(f2);

        // a
        F f3(f);
        butterfly2(f3);
        for (int i = 0; i < m; i++)
          f3[i] *= g2[i];
        butterfly_inv2(f3);
        f3.erase(f3.begin(), f3.begin() + m / 2);
        f3.resize(m, T(0)), butterfly2(f3);
        for (int i = 0; i < m; i++)
          f3[i] *= g2[i];
        butterfly_inv2(f3);
        for (int i = 0; i < m / 2; i++)
          f3[i] *= -im * im;
        g.insert(g.end(), f3.begin(), f3.begin() + m / 2);

        g2 = F(g), g2.resize(2 * m), butterfly2(g2);

        // b, c
        q = F(dh);
        q.resize(2 * m);
        for (int i = m - 1; i < 2 * m; i++)
          q[i] = T(0);
        butterfly2(q);
        for (int i = 0; i < 2 * m; i++)
          q[i] *= f2[i];
        butterfly_inv2(q);
        q = q.circular_mod(m);
        for (int i = 0; i < m; i++)
          q[i] *= i2m;

        // d, e
        q.resize(m + 1);
        s = ((f.differential() - q) << 1).circular_mod(m);
        s.resize(2 * m);
        butterfly2(s);
        for (int i = 0; i < 2 * m; i++)
          s[i] *= g2[i];
        butterfly_inv2(s);
        for (int i = 0; i < m; i++)
          s[i] *= i2m;
        s.resize(m);

        // f, g
        h = (*this);
        h.resize(2 * m), s.resize(2 * m);
        u = (h - (s << (m - 1)).integral()) >> m;
        butterfly2(u);
        for (int i = 0; i < 2 * m; i++)
          u[i] *= f2[i];
        butterfly_inv2(u);
        for (int i = 0; i < m; i++)
          u[i] *= i2m;
        u.resize(m);

        // h
        f.insert(f.end(), u.begin(), u.end());
      }
      return {f.begin(), f.begin() + n};
    }
    else
    //*/
    {
      F f{T(1)}, g{T(1)};
      while ((int)f.size() < n)
      {
        int m = f.size();
        g = convolution2(g, F{T(2)} - f * g, m);
        F q = (*this).differential();
        q.resize(m - 1);
        F r = f.convolution2(f, q).circular_mod(m);
        r.resize(m + 1);
        F s = ((f.differential() - r) << 1).circular_mod(m);
        F t = g * s;
        F h = (*this);
        h.resize(2 * m), t.resize(2 * m);
        F u = (h - (t << (m - 1)).integral()) >> m;
        F v = f * u;
        f.insert(f.end(), v.begin(), v.end());
      }
      return {f.begin(), f.begin() + n};
      /*
      F f{T(1)};
      while ((int)f.size() < n)
      {
        int m = f.size();
        f.resize(min(n, 2 * m), T(0));
        f *= (*this) + F{T(1)} - f.log();
      }
      return f;
      //*/
    }
  }

  F pow(const ll k) const
  {
    if (k == 0)
    {
      F res((*this).size(), T(0));
      res[0] = T(1);
      return res;
    }
    int n = (*this).size(), d;
    for (d = 0; d < n; d++)
    {
      if ((*this)[d] != T(0))
        break;
    }
    if (d == n)
      return F(n, 0);
    F res = F(*this) >> d;
    T c = res[0];
    res /= c;
    res = (res.log() * T(k)).exp();
    res *= c.pow(k), res <<= (d != 0 && k > n ? n : d * k);
    return res;
  }

  F powmod(ll k, const F &g) const
  {
    F res(2 * g.size(), 0);
    res.front() = 1;
    F tmp = (*this) % g;
    tmp.resize(g.size());
    while (k > 0)
    {
      if (k & 1)
      {
        res *= tmp;
        res %= g;
        res.resize(2 * g.size());
      }
      tmp = tmp.convolution2(tmp, tmp);
      tmp %= g;
      tmp.resize(g.size());
      k >>= 1;
    }
    return res;
  }

  // f(x)^k mod (x^n - 1)
  F powmod_circular(ll k, ll n) const
  {
    F res(n, 0);
    res.front() = 1;
    F tmp = (*this).circular_mod(n);
    while (k > 0)
    {
      if (k & 1)
        res = res.convolution2(res, tmp).circular_mod(n);
      tmp = tmp.convolution2(tmp, tmp).circular_mod(n);
      k >>= 1;
    }
    return res;
  }

  // 素数 mod を要求
  // 存在しないなら空配列を返す
  F sqrt() const
  {
    int n = (*this).size(), d;
    for (d = 0; d < n; d += 2)
    {
      if ((*this)[d] != 0)
        break;
      if (d + 1 < n && (*this)[d + 1] != 0)
        return F(0);
    }
    if (d >= n)
      return F(n, 0);

    T a = (*this)[d];
    int p = T::mod();
    int r = sqrt_mod(a.val(), p);
    if (r == -1)
      return F(0);

    T inv_2 = T(2).inv();
    F f = F(*this) >> d, res = F{r};
    while (res.size() < f.size())
    {
      res.resize(min(f.size(), 2 * res.size()), T(0));
      res = (res + res.inv() * f) * inv_2;
    }
    res <<= d / 2;
    return res;
  }

  F div_poly(const F &g) const
  {
    F f2 = F(*this), g2 = F(g);
    while (!f2.empty() && f2.back() == T(0))
      f2.pop_back();
    while (!g2.empty() && g2.back() == T(0))
      g2.pop_back();
    
    int n = f2.size(), m = g2.size();
    int k = n - m + 1;
    if (k <= 0)
      return F{};

    reverse(f2.begin(), f2.end());
    reverse(g2.begin(), g2.end());
    f2.resize(k, T(0)), g2.resize(k, T(0));
    F q = f2 / g2;
    reverse(q.begin(), q.end());
    while (!q.empty() && q.back() == T(0))
      q.pop_back();
    return q;
  }
  pair<F, F> divmod(const F &g) const
  {
    int m = g.size();
    assert(m != 0);
    F q = (*this).div_poly(g);

    F f3 = F(*this), g3 = F(g), q3 = F(q);
    f3.resize(m - 1, T(0)), g3.resize(m - 1, T(0)), q3.resize(m - 1, T(0));
    F r = f3 - q3 * g3;
    while (!r.empty() && r.back() == T(0))
      r.pop_back();

    return make_pair(q, r);
  }
  F operator%(const F &g) const { return (*this).divmod(g).second; }
  F &operator%=(const F &g) { return (*this) = (*this) % g; }

  F div_poly(const S &g) const
  {
    F f2 = F(*this);
    while (!f2.empty() && f2.back() == T(0))
      f2.pop_back();
    assert(!g.empty());

    int n = f2.size(), m = g.back().first + 1;
    int k = n - m + 1;
    if (k <= 0)
      return F{};

    reverse(f2.begin(), f2.end());
    S g2(g.size());
    for (int i = 0; i < (int)g.size(); i++)
      g2[(int)g.size() - 1 - i] = make_pair(m - 1 - g[i].first, g[i].second);
    f2.resize(k, T(0));
    F q = f2 / g2;
    reverse(q.begin(), q.end());
    while (!q.empty() && q.back() == T(0))
      q.pop_back();
    return q;
  }
  pair<F, F> divmod(const S &g) const
  {
    assert(!g.empty());
    int m = g.back().first + 1;
    F q = (*this).div_poly(g);

    F f3 = F(*this), q3 = F(q);
    f3.resize(m - 1, T(0)), q3.resize(m - 1, T(0));
    F r = f3 - q3 * g;
    while (!r.empty() && r.back() == T(0))
      r.pop_back();

    return make_pair(q, r);
  }
  F operator%(const S &g) const { return (*this).divmod(g).second; }
  F &operator%=(const S &g) { return (*this) = (*this) % g; }

  T eval(const T &x) const
  {
    T res(0);
    for (int i = (int)(*this).size() - 1; i >= 0; i--)
    {
      res *= x;
      res += (*this)[i];
    }
    return res;
  }

  // 各係数 a_n を n! で割る
  F to_egf()
  {
    for (int i = 0; i < (int)(*this).size(); i++)
      (*this)[i] *= get_finv(i);
    return (*this);
  }
  // 各係数 a_n を n! で割ったものを返す
  F get_egf() const { return F(*this).to_egf(); }
  // 各係数 a_n に n! をかける
  F to_ogf()
  {
    for (int i = 0; i < (int)(*this).size(); i++)
      (*this)[i] *= get_fac(i);
    return (*this);
  }
  // 各係数 a_n に n! をかけたものを返す
  F get_ogf() const { return F(*this).to_ogf(); }

  F taylor_shift(const T &c) const
  {
    int n = (*this).size();

    F f = F(*this).get_ogf();
    reverse(f.begin(), f.end());
    F g = F(n);
    g[0] = 1;
    for (int i = 1; i < n; i++)
      g[i] = c * g[i - 1];
    g.to_egf();
    F h = f * g;
    reverse(h.begin(), h.end());
    return h.to_egf();
  }

  vector<T> eval_multipoint(const vector<T> &xs) const
  {
    int m0 = xs.size(), m = 1;
    while (m < m0)
      m <<= 1;
    vector<F> node(2 * m, F{1});
    for (int i = 0; i < m0; i++)
      node[m + i] = {-xs[i], T(1)};
    for (int i = m - 1; i > 0; i--)
      node[i] = convolution2(node[i << 1], node[(i << 1) | 1]);

    node[1] = (*this).divmod(node[1]).second;
    for (int i = 2; i < m + m0; i++)
      node[i] = node[i >> 1].divmod(node[i]).second;

    vector<T> res(m0);
    for (int i = 0; i < m0; i++)
      res[i] = node[m + i].empty() ? T(0) : node[m + i][0];
    return res;
  }

  // i = 0..m-1 に対する f(ar^i)
  // https://noshi91.github.io/algorithm-encyclopedia/chirp-z-transform
  vector<T> eval_multipoint_geo(int m, T a, T r) const
  {
    if (r == 0)
    {
      vector<T> res(m, (*this).eval(0));
      res.front() = (*this).eval(a);
      return res;
    }

    auto calc_pw = [&](T x, int k) -> vector<T>
    {
      vector<T> res(k);
      res.front() = 1;
      for (int i = 1; i < k; i++)
        res[i] = res[i - 1] * x;
      return res;
    };
    auto get_pw_tri = [&](const vector<T> &pw, int i) -> T
    {
      if (i == 0)
        return 1;
      return i % 2 == 0 ? pw[i - 1].pow(i / 2) : pw[i].pow((i - 1) / 2); 
    };
    int n = (*this).size();
    T invr = r.inv();
    vector<T> pwa = calc_pw(a, n), pwr = calc_pw(r, n + m), pwir = calc_pw(invr, max(n, m));

    vector<T> s(n), t(n + m);
    for (int i = 0; i < n; i++)
      s[n - 1 - i] = (*this)[i] * pwa[i] * get_pw_tri(pwir, i);
    for (int i = 0; i < n + m; i++)
      t[i] = get_pw_tri(pwr, i);
    vector<T> u = convolution2(s, t, n + m - 1);
    u.erase(u.begin(), u.begin() + n - 1);
    for (int i = 0; i < m; i++)
      u[i] *= get_pw_tri(pwir, i);
    return u;
  }
};

// (次数, 係数) を昇順に並べたもの
template <class T, bool is_ntt_friendly>
struct SparseFormalPowerSeries : vector<pair<ll, T>>
{
  using vector<pair<ll, T>>::vector;
  using vector<pair<ll, T>>::operator=;
  using F = FormalPowerSeries<T, is_ntt_friendly>;
  using S = SparseFormalPowerSeries;

  F to_fps(int n) const
  {
    F res(n, T(0));
    for (auto [d, a] : (*this))
      res[d] += a;
    return res;
  }

  SparseFormalPowerSeries(const F &f)
  {
    (*this).clear();
    for (int i = 0; i < (int)f.size(); i++)
    {
      if (f[i] != T(0))
        (*this).emplace_back(make_pair(i, f[i]));
    }
  }

  S operator-() const
  {
    S res(*this);
    for (auto &[d, a] : res)
      a = -a;
    return res;
  }

  S operator*=(const T &k)
  {
    for (auto &[d, a] : (*this))
      a *= k;
    return (*this);
  }
  S operator/=(const T &k)
  {
    (*this) *= k.inv();
    return (*this);
  }
  S operator*(const T &k) const { return S(*this) *= k; }
  S operator/(const T &k) const { return S(*this) /= k; }
  friend S operator*(const T k, const S &f) { return f * k; }

  S operator+(const S &g) const
  {
    S res;
    int n = (*this).size(), m = g.size(), i = 0, j = 0;
    while (i < n || j < m)
    {
      pair<ll, T> tmp;
      if (j == m || (i != n && (*this)[i].first <= g[j].first))
        tmp = (*this)[i++];
      else
        tmp = g[j++];

      if (!res.empty() && res.back().first == tmp.first)
        res.back().second += tmp.second;
      else
        res.emplace_back(tmp);
    }
    return res;
  }
  S operator-(const S &g) const
  {
    S res;
    int n = (*this).size(), m = g.size(), i = 0, j = 0;
    while (i < n || j < m)
    {
      pair<ll, T> tmp;
      if (j == m || (i != n && (*this)[i].first <= g[j].first))
        tmp = (*this)[i++];
      else
      {
        tmp = g[j++];
        tmp.second = -tmp.second;
      }

      if (!res.empty() && res.back().first == tmp.first)
        res.back().second += tmp.second;
      else
        res.emplace_back(tmp);
    }
    return res;
  }
  S operator*(const S &g) const
  {
    S res;
    for (auto [d, a] : (*this))
      for (auto [e, b] : g)
        res.emplace_back(make_pair(d + e, a * b));
    sort(res.begin(), res.end(), [&](pair<ll, T> p1, pair<ll, T> p2)
         { return p1.first < p2.first; });
    S res2;
    for (auto da : res)
    {
      auto [d, a] = da;
      if (res2.empty() || res2.back().first != d)
        res2.emplace_back(da);
      else
        res2.back().second += a;
    }
    return res2;
  }
  S operator+=(const S &g) { return (*this) = (*this) + g; }
  S operator-=(const S &g) { return (*this) = (*this) - g; }
  S operator*=(const S &g) { return (*this) = (*this) * g; }

  S operator<<=(ll k)
  {
    for (auto &[d, a] : (*this))
      d += k;
    return (*this);
  }
  S operator<<(ll k) const { return (*this) <<= k; }
  S operator>>(ll k) const
  {
    S res;
    for (auto [d, a] : (*this))
    {
      d -= k;
      if (d >= 0)
        res.emplace_back(make_pair(d, a));
    }
    return res;
  }
  S operator>>=(ll k) { return (*this) = (*this) >> k; }

  F inv(int n) const
  {
    F f(n, T(0));
    f.front() = T(1);
    return f / (*this);
  }

  S differentiate()
  {
    for (auto &[d, a] : (*this))
      a *= d--;
    if (!(*this).empty() && (*this).front().first == -1)
      (*this).erase((*this).begin());
    return (*this);
  }
  S differential() const { return S(*this).differentiate(); }
  S integrate()
  {
    for (auto &[d, a] : (*this))
      a /= T(++d);
    return (*this);
  }
  S integral() const { return S(*this).integrate(); }

  F log(int n) const
  {
    F f = (*this).to_fps(n);
    return (f.differential() / (*this)).integral();
  }

  // 微分方程式 a(x)F'(x) + b(x)F(x) = 0, [x^0]F(x) = 1 を満たす F を n 項まで求める
  // [x^0]a(x) = 1 である必要がある
  F diffeq(const S &a, const S &b, int n) const
  {
    assert(a.front().first == 0 && a.front().second == 1);
    vector<T> minv(n);
    minv[1] = T(1);
    for (int i = 2; i < n; i++)
      minv[i] = -minv[T::mod() % i] * (T::mod() / i);
    F f(n, T(0));
    f[0] = T(1);
    for (int k = 0; k < n - 1; k++)
    {
      for (auto [i, ai] : a)
      {
        if (0 <= k - i + 1 && k - i + 1 < k + 1)
          f[k + 1] -= ai * (k - i + 1) * f[k - i + 1];
      }
      for (auto [j, bj] : b)
      {
        if (0 <= k - j && k - j < k + 1)
          f[k + 1] -= bj * f[k - j];
      }
      f[k + 1] *= minv[k + 1];
    }
    return f;
  }

  F exp(int n) const
  {
    return diffeq(S{{0, 1}}, -((*this).differential()), n);
  }
  
  // m >= 0 のときは O(nk)  (k: sparse の non-zero の個数)
  // m < 0 のときは O((n + d_0 m)k)
  F pow(ll m, int n) const
  {
    S f(*this);
    if (f.empty())
    {
      F res(n, T(0));
      if (m == 0)
        res.front() = T(1);
      return res;
    }
    auto [d0, a0] = f.front();
    T a0_inv = a0.inv();
    for (auto &[d, a] : f)
      d -= d0, a *= a0_inv;
    if (m >= 0)
    {
      F g = diffeq(f, -m * f.differential(), n);
      return (g * a0.pow(m)) << mul_limited(d0, m);
    }
    else
    {
      F g = diffeq(f, -m * f.differential(), n + (d0 * (-m)));
      F h = (g * a0_inv.pow(-m)) >> (d0 * (-m));
      h.resize(n);
      return h;
    }
  }

  // 素数 mod を要求
  // 存在しないなら空配列を返す
  F sqrt(int n) const
  {
    S f(*this);
    if (f.empty())
      return F(n, T(0));
    auto [d0, a0] = f.front();
    if (d0 % 2 != 0)
      return F(0);
    if (d0 >= n)
      return F(n, T(0));
    int p = T::mod();
    int r = sqrt_mod(a0.val(), p);
    if (r == -1)
      return F(0);

    T a0_inv = a0.inv();
    T inv_2 = T(2).inv();
    for (auto &[d, a] : f)
      d -= d0, a *= a0_inv;
    F g = diffeq(f, -inv_2 * f.differential(), n);
    return ((g * r) << (d0 / 2));
  }
};
template <class T, bool is_ntt_friendly>
vector<T> FormalPowerSeries<T, is_ntt_friendly>::fac{1, 1};
template <class T, bool is_ntt_friendly>
vector<T> FormalPowerSeries<T, is_ntt_friendly>::finv{1, 1};
template<class T, bool is_ntt_friendly>
vector<T> FormalPowerSeries<T, is_ntt_friendly>::invmint{0, 1};

template<class T, bool is_ntt_friendly>
struct RationalFormalPowerSeries
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;
  using R = RationalFormalPowerSeries;

  F num, den;

  R operator-() const
  {
    R res(*this);
    res.num = -res.num;
    return res;
  }

  R operator*=(const T &k)
  {
    (*this).num *= k;
    return *this;
  }
  R operator*(const T &k) const { return R(*this) *= k; }
  friend R operator*(const T k, const R &r) { return r * k; }
  R operator/=(const T &k)
  {
    (*this).den *= k;
    return k;
  }
  R operator/(const T &k) const { return R(*this) /= k; }

  R &operator+=(const R &r)
  {
    // ここうまくやると FFT の回数が節約できる気がする
    // うまくやらないと次数に偏りがある場合にかえって遅くなったりしそうで面倒
    F f, g;
    f = f.convolution2((*this).num, r.den);
    g = g.convolution2((*this).den, r.num);
    (*this).num = f + g;
    (*this).den = (*this).den.convolution2((*this).den, r.den);
    return *this;
  }
  R operator+(const R &r) const { return R(*this) += r; }
  R &operator-=(const R &r)
  {
    F f, g;
    f = f.convolution2((*this).num, r.den);
    g = g.convolution2((*this).den, r.num);
    (*this).num = f - g;
    (*this).den = (*this).den.convolution2((*this).den, r.den);
    return *this;
  }
  R operator-(const R &r) const { return R(*this) -= r; }
  
  R operator*=(const R &r)
  {
    (*this).num = (*this).num.convolution2((*this).num, r.num);
    (*this).den = (*this).den.convolution2((*this).den, r.den);
    return *this;
  }
  R operator*(const R &r) const { return R(*this) *= r; }
  R operator/=(const R &r)
  {
    (*this).num = (*this).num.convolution2((*this).num, r.den);
    (*this).den = (*this).den.convolution2((*this).den, r.num);
    return *this;
  }
  R operator/(const R &r) const { return R(*this) /= r; }

  R inv()
  {
    R res(*this);
    swap(res.num, res.den);
    return res;
  }
};

template <class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> convolution_many(const vector<FormalPowerSeries<T, is_ntt_friendly>> &fs, int d = -1)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  if (fs.empty())
  {
    if (d == -1)
      d = 0;
    F res(d + 1, T(0));
    res.front() = T(1);
    return res;
  }
  deque<F> deq;
  for (auto f : fs)
    deq.push_back(f);
  while ((int)deq.size() > 1)
  {
    F f = deq.front();
    deq.pop_front();
    F g = deq.front();
    deq.pop_front();
    f = f.convolution2(f, g, d);
    deq.push_back(f);
  }
  if (d != -1)
    deq.front().resize(d);
  return deq.front();
}

template <class T, bool is_ntt_friendly>
RationalFormalPowerSeries<T, is_ntt_friendly> rational_sum(const vector<RationalFormalPowerSeries<T, is_ntt_friendly>> &rs, int d = -1)
{
  using R = RationalFormalPowerSeries<T, is_ntt_friendly>;

  if (rs.empty())
    return R{{1}, {1}};

  deque<R> deq;
  for (auto &r : rs)
    deq.emplace_back(r);
  while ((int)deq.size() > 1)
  {
    R r1 = deq.front();
    deq.pop_front();
    R r2 = deq.front();
    deq.pop_front();
    R r3 = r1 + r2;
    if (d != -1)
    {
      if ((int)r3.num.size() > d)
        r3.num.resize(d);
      if ((int)r3.den.size() > d)
        r3.den.resize(d);
    }
    deq.emplace_back(r3);
  }
  if (d != -1)
    deq.front().num.resize(d), deq.front().den.resize(d);
  return deq.front();
}

template <class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> interpolation(const vector<T> &xs, const vector<T> &ys)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;
  using R = RationalFormalPowerSeries<T, is_ntt_friendly>;

  int n = xs.size();
  assert(n == ys.size());
  vector<F> fs(n);
  for (int i = 0; i < n; i++)
    fs[i] = F{-xs[i], T(1)};
  F g = convolution_many(fs);
  F h = g.differential();
  vector<T> a = h.eval_multipoint(xs);

  vector<R> rs(n);
  for (int i = 0; i < n; i++)
    rs[i] = R{F{ys[i] / a[i]}, fs[i]};
  R q = rational_sum(rs, n);
  return q.num;
}

// f(i) = ys[i] で定まる多項式 f(x) について f(c), …, f(c + M - 1) を求める
template<class T, bool is_ntt_friendly>
vector<T> sample_points_shift(const vector<T> &ys, int M, T c)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;
  F f;

  int N = ys.size();
  vector<T> a;
  {
    vector<T> p(N), q(N);
    for (int i = 0; i < N; i++)
    {
      p[i] = ys[i] * f.get_finv(i);
      q[i] = i % 2 == 0 ? f.get_finv(i) : -f.get_finv(i);
    }
    a = f.convolution2(p, q);
    a.resize(N);
  }
  vector<T> b;
  {
    vector<T> p(N), q(N);
    T tmp = 1;
    for (int i = 0; i < N; i++)
    {
      p[i] = a[i] * f.get_fac(i);
      q[i] = tmp * f.get_finv(i);
      tmp *= c - i;
    }
    reverse(q.begin(), q.end());
    b = f.convolution2(p, q);
    b.erase(b.begin(), b.begin() + N - 1);
    for (int i = 0; i < N; i++)
      b[i] *= f.get_finv(i);
  }
  vector<T> res;
  {
    vector<T> p(M);
    for (int i = 0; i < M; i++)
      p[i] = f.get_finv(i);
    res = f.convolution2(b, p);
    res.resize(M);
    for (int i = 0; i < M; i++)
      res[i] *= f.get_fac(i);
  }
  return res;
}

// https://suisen-kyopro.hatenablog.com/entry/2023/11/22/201600
// 前計算 O(K 2^K + (P/2^K) log K), クエリ O(2^K)
template<class T, bool is_ntt_friendly>
struct FactorialFast
{
private:
  const int P, K;
  vector<T> Y, Z, fac;

public:
  FactorialFast(const int K = 9) : P(T::mod()), K(K)
  {
    Y = {1};
    for (int i = 0; i < K; i++)
    {
      Z = sample_points_shift<T, is_ntt_friendly>(Y, (1 << (i + 2)) - (1 << i), 1 << i);
      Z.insert(Z.begin(), Y.begin(), Y.end());
      Y.resize(1 << (i + 1));
      for (int j = 0; j < (1 << (i + 1)); j++)
        Y[j] = Z[2 * j] * Z[2 * j + 1] * T::raw((1 << i) * (2 * j + 1));
    }
    if ((1 << K) <= P / (1 << K))
    {
      Z = sample_points_shift<T, is_ntt_friendly>(Y, P / (1 << K), 1 << K);
      Y.insert(Y.end(), Z.begin(), Z.end());
    }
    fac.resize(P / (1 << K) + 1);
    fac.at(0) = 1;
    for (int i = 0; i < P / (1 << K); i++)
      fac[i + 1] = fac[i] * Y[i] * T::raw((1 + i) * (1 << K));
  }
  T query(ll n)
  {
    if (n >= T::mod())
      return 0;
    T res = fac.at(n / (1 << K));
    for (int j = n / (1 << K) * (1 << K) + 1; j <= n; j++)
      res *= T::raw(j);
    return res;
  }
};

// f(x)/prod[i](1-a[i]x) = sum[i] c[i]/(1-a[i]x) なる c を求める
template<class T, bool is_ntt_friendly>
vector<T> partial_fraction_decomposition(const FormalPowerSeries<T, is_ntt_friendly> &f, const vector<T> &as)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  int N = as.size();
  assert((int)f.size() <= N);
  vector<T> bs(N);
  for (int i = 0; i < N; i++)
    bs[i] = as[i].inv();
  vector<F> gs(N);
  for (int i = 0; i < N; i++)
    gs[i] = F{T(1), -as[i]};
  F g = convolution_many(gs);
  F dg = g.differential();
  vector<T> ys = f.eval_multipoint(bs), zs = dg.eval_multipoint(bs);
  vector<T> cs(N);
  for (int i = 0; i < N; i++)
    cs[i] = -as[i] * ys[i] / zs[i];
  return cs;
}

// n = 0, 1, …, N-1 に対して n^k を列挙
template<class T>
vector<T> enum_pow(int N, int k)
{
  vector<int> minfactor(N, -1);
  for (int i = 2; i < N; i++)
  {
    if (minfactor[i] != -1)
      continue;
    for (int j = 2 * i; j < N; j += i)
      minfactor[j] = i;
  }
  vector<T> power(N);
  for (int i = 0; i < N; i++)
  {
    if (minfactor[i] == -1)
      power[i] = T(i).pow(k);
    else
      power[i] = power[minfactor[i]] * power[i / minfactor[i]];
  }
  return power;
}

// sum_[i = 0..infty] r^i i^d
template <class T, bool is_ntt_friendly>
T sum_of_exp_times_poly_limit(T r, int d)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  vector<T> pws = enum_pow<T>(d + 2, d);
  vector<T> pwr(d + 2, 1);
  for (int i = 0; i < d + 1; i++)
    pwr[i + 1] = pwr[i] * r;
  F f(d + 2), g(d + 2);
  f.front() = pws.front() * pwr.front();
  for (int i = 0; i < d + 1; i++)
    f[i + 1] = f[i] + pws[i + 1] * pwr[i + 1];
  for (int i = 0; i <= d + 1; i++)
    g[i] = (i % 2 == 0 ? 1 : -1) * pwr[i] * g.get_fac(d + 1) * g.get_finv(i) * g.get_finv(d + 1 - i);
  T c = 0;
  for (int i = 0; i <= d + 1; i++)
    c += f[i] * g[d + 1 - i];
  c /= accumulate(g.begin(), g.end(), T(0));
  return c;
}

// sum_[i = 0..n-1] r^i i^d
template <class T, bool is_ntt_friendly>
T sum_of_exp_times_poly(T r, int d, ll n)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  if (n == 0)
    return 0;
  if (r == 0)
    return d == 0 ? 1 : 0;

  vector<T> pws = enum_pow<T>(d + 2, d);
  vector<T> pwr(d + 2, 1);
  for (int i = 0; i < d + 1; i++)
    pwr[i + 1] = pwr[i] * r;
  F f(d + 2), g(d + 2);
  f.front() = pws.front() * pwr.front();
  for (int i = 0; i < d + 1; i++)
    f[i + 1] = f[i] + pws[i + 1] * pwr[i + 1];
  if (r == 1)
  {
    LagrangeInterpolation<T> lag(f);
    return lag.eval(n - 1);
  }
  for (int i = 0; i <= d + 1; i++)
    g[i] = (i % 2 == 0 ? 1 : -1) * pwr[i] * g.get_fac(d + 1) * g.get_finv(i) * g.get_finv(d + 1 - i);
  T c = 0;
  for (int i = 0; i <= d + 1; i++)
    c += f[i] * g[d + 1 - i];
  c /= accumulate(g.begin(), g.end(), T(0));
  F h = f - F(d + 2, c);
  {
    T rinv = r.inv();
    T pwrinv = 1;
    for (int i = 0; i <= d + 1; i++)
    {
      h[i] *= pwrinv;
      pwrinv *= rinv;
    }
  }
  LagrangeInterpolation<T> lag(h);
  return c + r.pow(n - 1) * lag.eval(n - 1);
}

// prod[d in D](1 + cx^d) を M 次の項まで求める
template <class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> multiply_many(const int &M, const T &c, const vector<int> &D)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  vector<int> cnt(M + 1, 0);
  for (auto d : D)
  {
    if (d < 0 || M < d)
      continue;
    cnt[d]++;
  }

  F f(M + 1, 0);
  for (int k = 1; k <= M; k++)
  {
    T pw = 1;
    for (int i = 1; k * i <= M; i++)
    {
      pw *= c;
      if (i & 1)
        f[k * i] += T::raw(cnt[k]) * pw * f.get_invmint(i);
      else
        f[k * i] -= T::raw(cnt[k]) * pw * f.get_invmint(i);
    }
  }
  return f.exp();
}

// 多重集合 S の要素から何個か選んで総和を 0, 1, …, M にする方法の数
template <class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> subset_sum(const int &M, const vector<int> &S)
{
  return multiply_many<T, is_ntt_friendly>(M, T(1), S);
}

// 集合 S の各要素が無限個ある集合 T から何個か選んで総和を 0, 1, …, M にする方法の数
template <class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> partition(const int &M, const vector<int> &S)
{
  return multiply_many<T, is_ntt_friendly>(M, T(-1), S).inv();
}

template<class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> stirling1(const int &N)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;
  using S = SparseFormalPowerSeries<T, is_ntt_friendly>;

  if (N == 0)
    return {1};
  if (N == 1)
    return {0, 1};
  if (N & 1)
  {
    F f = stirling1<T, is_ntt_friendly>(N - 1);
    f.resize(N + 1, T(0));
    return f * S{{0, 1 - N}, {1, 1}};
  }
  else
  {
    F f = stirling1<T, is_ntt_friendly>(N / 2);
    f.resize(N + 1, T(0));
    F g = f.taylor_shift(-(N / 2));
    return f * g;
  }
}

template<class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> stirling2(const int &N)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  vector<T> power = enum_pow<T>(N + 1, N);
  F A(N + 1), B(N + 1);
  for (int i = 0; i <= N; i++)
  {
    A[i] = power[i] * A.get_finv(i);
    B[i] = (i & 1) ? -A.get_finv(i) : A.get_finv(i);
  }
  return A * B;
}

template<class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> bernoulli_number(const int &N)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  F f(N + 1, T(0));
  for (int i = 0; i <= N; i++)
    f[i] = f.get_finv(i + 1);
  return f.inv().to_ogf();
}

// [x^N] P(x)/Q(x) を求める（P の次数は Q の次数より小さい）
template<class T, bool is_ntt_friendly>
T bostan_mori(const FormalPowerSeries<T, is_ntt_friendly> &P, const FormalPowerSeries<T, is_ntt_friendly> &Q, ll N)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  int d = (int)Q.size() - 1;
  assert((int)P.size() <= d);
  if (is_ntt_friendly)
  {
    int z = 1;
    while (z < 2 * d + 1)
      z <<= 1;
    T iz = T(z).inv();
    F U = F(P), V = F(Q);
    U.resize(z), V.resize(z);
    while (N > 0)
    {
      U.butterfly2(U), V.butterfly2(V);
      for (int i = 0; i < z; i += 2)
      {
        T x = V[i + 1], y = V[i];
        U[i] *= x, V[i] *= x;
        U[i + 1] *= y, V[i + 1] *= y;
      }
      U.butterfly_inv2(U), V.butterfly_inv2(V);
      for (int i = 0; i < (z >> 1); i++)
      {
        U[i] = U[2 * i + (N & 1)] * iz;
        V[i] = V[2 * i] * iz;
      }
      for (int i = (z >> 1); i < z; i++)
        U[i] = 0, V[i] = 0;
      N >>= 1;
    }
    return U.front() / V.front();
  }
  else
  {
    F U = F(P), V = F(Q);
    U.resize(d), V.resize(d + 1);
    while (N > 0)
    {
      F U2 = F(U), V2 = F(V), V3 = F(V);
      for (int i = 1; i <= d; i += 2)
        V3[i] = -V3[i];
      U2 *= V3, V2 *= V3;
      for (int i = 0; i <= d; i++)
      {
        U[i] = U2[2 * i + (N & 1)];
        V[i] = V2[2 * i];
      }
      N >>= 1;
    }
    return U.front() / V.front();
  }
}
// a_n = sum[i = 1..d] c_i a_{n-i}（n ≥ d）を満たすとき、a_N を求める（A は 0-indexed で C は 1-indexed）
template<class T, bool is_ntt_friendly>
T linear_recurrence(const vector<T> &A, const vector<T> &C, ll N)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  int d = C.size();
  assert((int)A.size() >= d);

  F Ga(d), Q(d + 1);
  Q[0] = 1;
  for (int i = 0; i < d; i++)
    Ga[i] = A[i], Q[i + 1] = -C[i];
  F P = Ga * Q;
  return bostan_mori(P, Q, N);
}

// (P の次数) < (Q の次数) とする
// P/Q = R + x^N (P'/Q) を満たす P' (R は N 次未満、P' は d 次未満)
// [x^{N+n}](P/Q) = [x^n](P'/Q)　線形漸化的数列のシフト
// 高速化の余地あり
template<class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> fiduccia(const FormalPowerSeries<T, is_ntt_friendly> &P, const FormalPowerSeries<T, is_ntt_friendly> &Q, ll N)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  assert(P.size() < Q.size());

  F xinv = -(Q >> 1);
  if (Q[0] != 1)
    xinv /= Q[0];
  return xinv.powmod(N, Q) * P % Q;
}
// a_n = sum[i = 1..d] c_i a_{n-i}（n ≥ d）を満たすとき、a_N, …, a_{N+d-1} を求める（A は 0-indexed で C は 1-indexed）
template<class T, bool is_ntt_friendly>
vector<T> linear_recurrence_many(const vector<T> &A, const vector<T> &C, ll N)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;

  int d = C.size();
  assert((int)A.size() >= d);

  F Ga(d), Q(d + 1);
  Q[0] = 1;
  for (int i = 0; i < d; i++)
    Ga[i] = A[i], Q[i + 1] = -C[i];
  F P = Ga * Q;
  F P2 = fiduccia(P, Q, N);
  P2.resize(d);
  F Gb = P2 / Q + (P.div_poly(Q) >> N);
  return Gb;
}

// Σ[i = 0..M-1] a_i exp(b_i x) を N 項まで求める
template<class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> exp_sum(int N, const vector<T> &a, const vector<T> &b)
{
  using F = FormalPowerSeries<T, is_ntt_friendly>;
  using R = RationalFormalPowerSeries<T, is_ntt_friendly>;
  assert(a.size() == b.size());
  int M = a.size();
  vector<R> gs(M);
  for (int i = 0; i < M; i++)
    gs[i] = R{F{a[i]}, F{1, -b[i]}};
  R g = rational_sum(gs, N);
  return (g.num / g.den).to_egf();
}

// f(exp(kx)) を N 項まで求める 
template<class T, bool is_ntt_friendly>
FormalPowerSeries<T, is_ntt_friendly> eval_exp(FormalPowerSeries<T, is_ntt_friendly> &f, T k, int N = -1)
{
  if (N == -1)
    N = (int)f.size();
  vector<T> b(f.size());
  for (int i = 0; i < (int)f.size(); i++)
    b[i] = k * i;
  return exp_sum<T, is_ntt_friendly>(N, f, b);
}

/*
using mint = modint998244353;
const bool ntt = true;
//*/
//*
using mint = modint1000000007;
const bool ntt = false;
//*/
/*
using mint = modint;
const bool ntt = false;
//*/

using fps = FormalPowerSeries<mint, ntt>;
using sfps = SparseFormalPowerSeries<mint, ntt>;
using rfps = RationalFormalPowerSeries<mint, ntt>;

int main()
{
  ll N;
  cin >> N;
  FactorialFast<mint, ntt> fac(13);
  cout << fac.query(N).val() << endl;
}