/**
 * date   : 2025-01-04 18:13:27
 * author : Nyaan
 */

#define NDEBUG

using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tr2/dynamic_bitset>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// utility

namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;

template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;

template <typename T, typename U>
struct P : pair<T, U> {
  template <typename... Args>
  constexpr P(Args... args) : pair<T, U>(args...) {}

  using pair<T, U>::first;
  using pair<T, U>::second;

  P &operator+=(const P &r) {
    first += r.first;
    second += r.second;
    return *this;
  }
  P &operator-=(const P &r) {
    first -= r.first;
    second -= r.second;
    return *this;
  }
  P &operator*=(const P &r) {
    first *= r.first;
    second *= r.second;
    return *this;
  }
  template <typename S>
  P &operator*=(const S &r) {
    first *= r, second *= r;
    return *this;
  }
  P operator+(const P &r) const { return P(*this) += r; }
  P operator-(const P &r) const { return P(*this) -= r; }
  P operator*(const P &r) const { return P(*this) *= r; }
  template <typename S>
  P operator*(const S &r) const {
    return P(*this) *= r;
  }
  P operator-() const { return P{-first, -second}; }
};

using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;

constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;

template <typename T>
int sz(const T &t) {
  return t.size();
}

template <typename T, typename U>
inline bool amin(T &x, U y) {
  return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
  return (x < y) ? (x = y, true) : false;
}

template <typename T>
inline T Max(const vector<T> &v) {
  return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
  return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
  return accumulate(begin(v), end(v), 0LL);
}

template <typename T>
int lb(const vector<T> &v, const T &a) {
  return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
  return upper_bound(begin(v), end(v), a) - begin(v);
}

constexpr long long TEN(int n) {
  long long ret = 1, x = 10;
  for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
  return ret;
}

template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
  return make_pair(t, u);
}

template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
  vector<T> ret(v.size() + 1);
  if (rev) {
    for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
  } else {
    for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
  }
  return ret;
};

template <typename T>
vector<T> mkuni(const vector<T> &v) {
  vector<T> ret(v);
  sort(ret.begin(), ret.end());
  ret.erase(unique(ret.begin(), ret.end()), ret.end());
  return ret;
}

template <typename F>
vector<int> mkord(int N, F f) {
  vector<int> ord(N);
  iota(begin(ord), end(ord), 0);
  sort(begin(ord), end(ord), f);
  return ord;
}

template <typename T>
vector<int> mkinv(vector<T> &v) {
  int max_val = *max_element(begin(v), end(v));
  vector<int> inv(max_val + 1, -1);
  for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
  return inv;
}

vector<int> mkiota(int n) {
  vector<int> ret(n);
  iota(begin(ret), end(ret), 0);
  return ret;
}

template <typename T>
T mkrev(const T &v) {
  T w{v};
  reverse(begin(w), end(w));
  return w;
}

template <typename T>
bool nxp(T &v) {
  return next_permutation(begin(v), end(v));
}

// 返り値の型は入力の T に依存
// i 要素目 : [0, a[i])
template <typename T>
vector<vector<T>> product(const vector<T> &a) {
  vector<vector<T>> ret;
  vector<T> v;
  auto dfs = [&](auto rc, int i) -> void {
    if (i == (int)a.size()) {
      ret.push_back(v);
      return;
    }
    for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back();
  };
  dfs(dfs, 0);
  return ret;
}

// F : function(void(T&)), mod を取る操作
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I, const function<void(T &)> &f) {
  T res = I;
  for (; n; f(a = a * a), n >>= 1) {
    if (n & 1) f(res = res * a);
  }
  return res;
}
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I = T{1}) {
  return Power(a, n, I, function<void(T &)>{[](T &) -> void {}});
}

template <typename T>
T Rev(const T &v) {
  T res = v;
  reverse(begin(res), end(res));
  return res;
}

template <typename T>
vector<T> Transpose(const vector<T> &v) {
  using U = typename T::value_type;
  if(v.empty()) return {};
  int H = v.size(), W = v[0].size();
  vector res(W, T(H, U{}));
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) {
      res[j][i] = v[i][j];
    }
  }
  return res;
}

template <typename T>
vector<T> Rotate(const vector<T> &v, int clockwise = true) {
  using U = typename T::value_type;
  int H = v.size(), W = v[0].size();
  vector res(W, T(H, U{}));
  for (int i = 0; i < H; i++) {
    for (int j = 0; j < W; j++) {
      if (clockwise) {
        res[W - 1 - j][i] = v[i][j];
      } else {
        res[j][H - 1 - i] = v[i][j];
      }
    }
  }
  return res;
}

}  // namespace Nyaan


// bit operation

namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
  return __builtin_popcountll(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
  return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
  if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
}  // namespace Nyaan


// inout

namespace Nyaan {

template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
  os << p.first << " " << p.second;
  return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
  is >> p.first >> p.second;
  return is;
}

template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
  int s = (int)v.size();
  for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
  return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
  for (auto &x : v) is >> x;
  return is;
}

istream &operator>>(istream &is, __int128_t &x) {
  string S;
  is >> S;
  x = 0;
  int flag = 0;
  for (auto &c : S) {
    if (c == '-') {
      flag = true;
      continue;
    }
    x *= 10;
    x += c - '0';
  }
  if (flag) x = -x;
  return is;
}

istream &operator>>(istream &is, __uint128_t &x) {
  string S;
  is >> S;
  x = 0;
  for (auto &c : S) {
    x *= 10;
    x += c - '0';
  }
  return is;
}

ostream &operator<<(ostream &os, __int128_t x) {
  if (x == 0) return os << 0;
  if (x < 0) os << '-', x = -x;
  string S;
  while (x) S.push_back('0' + x % 10), x /= 10;
  reverse(begin(S), end(S));
  return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
  if (x == 0) return os << 0;
  string S;
  while (x) S.push_back('0' + x % 10), x /= 10;
  reverse(begin(S), end(S));
  return os << S;
}

void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
  cin >> t;
  in(u...);
}

void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
  cout << t;
  if (sizeof...(u)) cout << sep;
  out(u...);
}

struct IoSetupNya {
  IoSetupNya() {
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    cout << fixed << setprecision(15);
    cerr << fixed << setprecision(7);
  }
} iosetupnya;

}  // namespace Nyaan


// debug


#ifdef NyaanDebug
#define trc(...) (void(0))
#else
#define trc(...) (void(0))
#endif

#ifdef NyaanLocal
#define trc2(...) (void(0))
#else
#define trc2(...) (void(0))
#endif


// macro

#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...)   \
  int __VA_ARGS__; \
  in(__VA_ARGS__)
#define inl(...)         \
  long long __VA_ARGS__; \
  in(__VA_ARGS__)
#define ins(...)      \
  string __VA_ARGS__; \
  in(__VA_ARGS__)
#define in2(s, t)                           \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i]);                         \
  }
#define in3(s, t, u)                        \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i], u[i]);                   \
  }
#define in4(s, t, u, v)                     \
  for (int i = 0; i < (int)s.size(); i++) { \
    in(s[i], t[i], u[i], v[i]);             \
  }
#define die(...)             \
  do {                       \
    Nyaan::out(__VA_ARGS__); \
    return;                  \
  } while (0)


namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }


//



using namespace std;




using namespace std;

namespace internal {
template <typename T>
using is_broadly_integral =
    typename conditional_t<is_integral_v<T> || is_same_v<T, __int128_t> ||
                               is_same_v<T, __uint128_t>,
                           true_type, false_type>::type;

template <typename T>
using is_broadly_signed =
    typename conditional_t<is_signed_v<T> || is_same_v<T, __int128_t>,
                           true_type, false_type>::type;

template <typename T>
using is_broadly_unsigned =
    typename conditional_t<is_unsigned_v<T> || is_same_v<T, __uint128_t>,
                           true_type, false_type>::type;

#define ENABLE_VALUE(x) \
  template <typename T> \
  constexpr bool x##_v = x<T>::value;

ENABLE_VALUE(is_broadly_integral);
ENABLE_VALUE(is_broadly_signed);
ENABLE_VALUE(is_broadly_unsigned);
#undef ENABLE_VALUE

#define ENABLE_HAS_TYPE(var)                                   \
  template <class, class = void>                               \
  struct has_##var : false_type {};                            \
  template <class T>                                           \
  struct has_##var<T, void_t<typename T::var>> : true_type {}; \
  template <class T>                                           \
  constexpr auto has_##var##_v = has_##var<T>::value;

#define ENABLE_HAS_VAR(var)                                     \
  template <class, class = void>                                \
  struct has_##var : false_type {};                             \
  template <class T>                                            \
  struct has_##var<T, void_t<decltype(T::var)>> : true_type {}; \
  template <class T>                                            \
  constexpr auto has_##var##_v = has_##var<T>::value;

}  // namespace internal




template <uint32_t mod>
struct LazyMontgomeryModInt {
  using mint = LazyMontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 ret = mod;
    for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
    return ret;
  }

  static constexpr u32 r = get_r();
  static constexpr u32 n2 = -u64(mod) % mod;
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
  static_assert(r * mod == 1, "this code has bugs.");

  u32 a;

  constexpr LazyMontgomeryModInt() : a(0) {}
  constexpr LazyMontgomeryModInt(const int64_t &b)
      : a(reduce(u64(b % mod + mod) * n2)){};

  static constexpr u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
  }

  constexpr mint &operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

  constexpr mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
  constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
  constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
  constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
  constexpr bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr mint operator-() const { return mint() - mint(*this); }
  constexpr mint operator+() const { return mint(*this); }

  constexpr mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  constexpr mint inverse() const {
    int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
    while (y > 0) {
      t = x / y;
      x -= t * y, u -= t * v;
      tmp = x, x = y, y = tmp;
      tmp = u, u = v, v = tmp;
    }
    return mint{u};
  }

  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }

  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = LazyMontgomeryModInt<mod>(t);
    return (is);
  }

  constexpr u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }

  static constexpr u32 get_mod() { return mod; }
};






__attribute__((target("sse4.2"))) inline __m128i my128_mullo_epu32(
    const __m128i &a, const __m128i &b) {
  return _mm_mullo_epi32(a, b);
}

__attribute__((target("sse4.2"))) inline __m128i my128_mulhi_epu32(
    const __m128i &a, const __m128i &b) {
  __m128i a13 = _mm_shuffle_epi32(a, 0xF5);
  __m128i b13 = _mm_shuffle_epi32(b, 0xF5);
  __m128i prod02 = _mm_mul_epu32(a, b);
  __m128i prod13 = _mm_mul_epu32(a13, b13);
  __m128i prod = _mm_unpackhi_epi64(_mm_unpacklo_epi32(prod02, prod13),
                                    _mm_unpackhi_epi32(prod02, prod13));
  return prod;
}

__attribute__((target("sse4.2"))) inline __m128i montgomery_mul_128(
    const __m128i &a, const __m128i &b, const __m128i &r, const __m128i &m1) {
  return _mm_sub_epi32(
      _mm_add_epi32(my128_mulhi_epu32(a, b), m1),
      my128_mulhi_epu32(my128_mullo_epu32(my128_mullo_epu32(a, b), r), m1));
}

__attribute__((target("sse4.2"))) inline __m128i montgomery_add_128(
    const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
  __m128i ret = _mm_sub_epi32(_mm_add_epi32(a, b), m2);
  return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}

__attribute__((target("sse4.2"))) inline __m128i montgomery_sub_128(
    const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
  __m128i ret = _mm_sub_epi32(a, b);
  return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}

__attribute__((target("avx2"))) inline __m256i my256_mullo_epu32(
    const __m256i &a, const __m256i &b) {
  return _mm256_mullo_epi32(a, b);
}

__attribute__((target("avx2"))) inline __m256i my256_mulhi_epu32(
    const __m256i &a, const __m256i &b) {
  __m256i a13 = _mm256_shuffle_epi32(a, 0xF5);
  __m256i b13 = _mm256_shuffle_epi32(b, 0xF5);
  __m256i prod02 = _mm256_mul_epu32(a, b);
  __m256i prod13 = _mm256_mul_epu32(a13, b13);
  __m256i prod = _mm256_unpackhi_epi64(_mm256_unpacklo_epi32(prod02, prod13),
                                       _mm256_unpackhi_epi32(prod02, prod13));
  return prod;
}

__attribute__((target("avx2"))) inline __m256i montgomery_mul_256(
    const __m256i &a, const __m256i &b, const __m256i &r, const __m256i &m1) {
  return _mm256_sub_epi32(
      _mm256_add_epi32(my256_mulhi_epu32(a, b), m1),
      my256_mulhi_epu32(my256_mullo_epu32(my256_mullo_epu32(a, b), r), m1));
}

__attribute__((target("avx2"))) inline __m256i montgomery_add_256(
    const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
  __m256i ret = _mm256_sub_epi32(_mm256_add_epi32(a, b), m2);
  return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
                          ret);
}

__attribute__((target("avx2"))) inline __m256i montgomery_sub_256(
    const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
  __m256i ret = _mm256_sub_epi32(a, b);
  return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
                          ret);
}

namespace ntt_inner {
using u64 = uint64_t;
constexpr uint32_t get_pr(uint32_t mod) {
  if (mod == 2) return 1;
  u64 ds[32] = {};
  int idx = 0;
  u64 m = mod - 1;
  for (u64 i = 2; i * i <= m; ++i) {
    if (m % i == 0) {
      ds[idx++] = i;
      while (m % i == 0) m /= i;
    }
  }
  if (m != 1) ds[idx++] = m;

  uint32_t pr = 2;
  while (1) {
    int flg = 1;
    for (int i = 0; i < idx; ++i) {
      u64 a = pr, b = (mod - 1) / ds[i], r = 1;
      while (b) {
        if (b & 1) r = r * a % mod;
        a = a * a % mod;
        b >>= 1;
      }
      if (r == 1) {
        flg = 0;
        break;
      }
    }
    if (flg == 1) break;
    ++pr;
  }
  return pr;
}

constexpr int SZ_FFT_BUF = 1 << 23;
uint32_t _buf1[SZ_FFT_BUF] __attribute__((aligned(64)));
uint32_t _buf2[SZ_FFT_BUF] __attribute__((aligned(64)));
}  // namespace ntt_inner

template <typename mint>
struct NTT {
  static constexpr uint32_t mod = mint::get_mod();
  static constexpr uint32_t pr = ntt_inner::get_pr(mint::get_mod());
  static constexpr int level = __builtin_ctzll(mod - 1);
  mint dw[level], dy[level];
  mint *buf1, *buf2;

  constexpr NTT() {
    setwy(level);
    union raw_cast {
      mint dat;
      uint32_t _;
    };
    buf1 = &(((raw_cast *)(ntt_inner::_buf1))->dat);
    buf2 = &(((raw_cast *)(ntt_inner::_buf2))->dat);
  }

  constexpr void setwy(int k) {
    mint w[level], y[level];
    w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
    y[k - 1] = w[k - 1].inverse();
    for (int i = k - 2; i > 0; --i)
      w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
    dw[0] = dy[0] = w[1] * w[1];
    dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
    for (int i = 3; i < k; ++i) {
      dw[i] = dw[i - 1] * y[i - 2] * w[i];
      dy[i] = dy[i - 1] * w[i - 2] * y[i];
    }
  }

  __attribute__((target("avx2"))) void ntt(mint *a, int n) {
    int k = n ? __builtin_ctz(n) : 0;
    if (k == 0) return;
    if (k == 1) {
      mint a1 = a[1];
      a[1] = a[0] - a[1];
      a[0] = a[0] + a1;
      return;
    }
    if (k & 1) {
      int v = 1 << (k - 1);
      if (v < 8) {
        for (int j = 0; j < v; ++j) {
          mint ajv = a[j + v];
          a[j + v] = a[j] - ajv;
          a[j] += ajv;
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        int j0 = 0;
        int j1 = v;
        for (; j0 < v; j0 += 8, j1 += 8) {
          __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
          __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
          __m256i naj = montgomery_add_256(T0, T1, m2, m0);
          __m256i najv = montgomery_sub_256(T0, T1, m2, m0);
          _mm256_storeu_si256((__m256i *)(a + j0), naj);
          _mm256_storeu_si256((__m256i *)(a + j1), najv);
        }
      }
    }
    int u = 1 << (2 + (k & 1));
    int v = 1 << (k - 2 - (k & 1));
    mint one = mint(1);
    mint imag = dw[1];
    while (v) {
      if (v == 1) {
        mint ww = one, xx = one, wx = one;
        for (int jh = 0; jh < u;) {
          ww = xx * xx, wx = ww * xx;
          mint t0 = a[jh + 0], t1 = a[jh + 1] * xx;
          mint t2 = a[jh + 2] * ww, t3 = a[jh + 3] * wx;
          mint t0p2 = t0 + t2, t1p3 = t1 + t3;
          mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
          a[jh + 0] = t0p2 + t1p3, a[jh + 1] = t0p2 - t1p3;
          a[jh + 2] = t0m2 + t1m3, a[jh + 3] = t0m2 - t1m3;
          xx *= dw[__builtin_ctz((jh += 4))];
        }
      } else if (v == 4) {
        const __m128i m0 = _mm_set1_epi32(0);
        const __m128i m1 = _mm_set1_epi32(mod);
        const __m128i m2 = _mm_set1_epi32(mod + mod);
        const __m128i r = _mm_set1_epi32(mint::r);
        const __m128i Imag = _mm_set1_epi32(imag.a);
        mint ww = one, xx = one, wx = one;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = v;
            for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i T0P2 = montgomery_add_128(T0, T2, m2, m0);
              const __m128i T1P3 = montgomery_add_128(T1, T3, m2, m0);
              const __m128i T0M2 = montgomery_sub_128(T0, T2, m2, m0);
              const __m128i T1M3 = montgomery_mul_128(
                  montgomery_sub_128(T1, T3, m2, m0), Imag, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_sub_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j2),
                               montgomery_add_128(T0M2, T1M3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j3),
                               montgomery_sub_128(T0M2, T1M3, m2, m0));
            }
          } else {
            ww = xx * xx, wx = ww * xx;
            const __m128i WW = _mm_set1_epi32(ww.a);
            const __m128i WX = _mm_set1_epi32(wx.a);
            const __m128i XX = _mm_set1_epi32(xx.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i MT1 = montgomery_mul_128(T1, XX, r, m1);
              const __m128i MT2 = montgomery_mul_128(T2, WW, r, m1);
              const __m128i MT3 = montgomery_mul_128(T3, WX, r, m1);
              const __m128i T0P2 = montgomery_add_128(T0, MT2, m2, m0);
              const __m128i T1P3 = montgomery_add_128(MT1, MT3, m2, m0);
              const __m128i T0M2 = montgomery_sub_128(T0, MT2, m2, m0);
              const __m128i T1M3 = montgomery_mul_128(
                  montgomery_sub_128(MT1, MT3, m2, m0), Imag, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_sub_128(T0P2, T1P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j2),
                               montgomery_add_128(T0M2, T1M3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j3),
                               montgomery_sub_128(T0M2, T1M3, m2, m0));
            }
          }
          xx *= dw[__builtin_ctz((jh += 4))];
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m1 = _mm256_set1_epi32(mod);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        const __m256i r = _mm256_set1_epi32(mint::r);
        const __m256i Imag = _mm256_set1_epi32(imag.a);
        mint ww = one, xx = one, wx = one;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = v;
            for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i T0P2 = montgomery_add_256(T0, T2, m2, m0);
              const __m256i T1P3 = montgomery_add_256(T1, T3, m2, m0);
              const __m256i T0M2 = montgomery_sub_256(T0, T2, m2, m0);
              const __m256i T1M3 = montgomery_mul_256(
                  montgomery_sub_256(T1, T3, m2, m0), Imag, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_sub_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j2),
                                  montgomery_add_256(T0M2, T1M3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j3),
                                  montgomery_sub_256(T0M2, T1M3, m2, m0));
            }
          } else {
            ww = xx * xx, wx = ww * xx;
            const __m256i WW = _mm256_set1_epi32(ww.a);
            const __m256i WX = _mm256_set1_epi32(wx.a);
            const __m256i XX = _mm256_set1_epi32(xx.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i MT1 = montgomery_mul_256(T1, XX, r, m1);
              const __m256i MT2 = montgomery_mul_256(T2, WW, r, m1);
              const __m256i MT3 = montgomery_mul_256(T3, WX, r, m1);
              const __m256i T0P2 = montgomery_add_256(T0, MT2, m2, m0);
              const __m256i T1P3 = montgomery_add_256(MT1, MT3, m2, m0);
              const __m256i T0M2 = montgomery_sub_256(T0, MT2, m2, m0);
              const __m256i T1M3 = montgomery_mul_256(
                  montgomery_sub_256(MT1, MT3, m2, m0), Imag, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_sub_256(T0P2, T1P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j2),
                                  montgomery_add_256(T0M2, T1M3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j3),
                                  montgomery_sub_256(T0M2, T1M3, m2, m0));
            }
          }
          xx *= dw[__builtin_ctz((jh += 4))];
        }
      }
      u <<= 2;
      v >>= 2;
    }
  }

  __attribute__((target("avx2"))) void intt(mint *a, int n,
                                            int normalize = true) {
    int k = n ? __builtin_ctz(n) : 0;
    if (k == 0) return;
    if (k == 1) {
      mint a1 = a[1];
      a[1] = a[0] - a[1];
      a[0] = a[0] + a1;
      if (normalize) {
        a[0] *= mint(2).inverse();
        a[1] *= mint(2).inverse();
      }
      return;
    }
    int u = 1 << (k - 2);
    int v = 1;
    mint one = mint(1);
    mint imag = dy[1];
    while (u) {
      if (v == 1) {
        mint ww = one, xx = one, yy = one;
        u <<= 2;
        for (int jh = 0; jh < u;) {
          ww = xx * xx, yy = xx * imag;
          mint t0 = a[jh + 0], t1 = a[jh + 1];
          mint t2 = a[jh + 2], t3 = a[jh + 3];
          mint t0p1 = t0 + t1, t2p3 = t2 + t3;
          mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
          a[jh + 0] = t0p1 + t2p3, a[jh + 2] = (t0p1 - t2p3) * ww;
          a[jh + 1] = t0m1 + t2m3, a[jh + 3] = (t0m1 - t2m3) * ww;
          xx *= dy[__builtin_ctz(jh += 4)];
        }
      } else if (v == 4) {
        const __m128i m0 = _mm_set1_epi32(0);
        const __m128i m1 = _mm_set1_epi32(mod);
        const __m128i m2 = _mm_set1_epi32(mod + mod);
        const __m128i r = _mm_set1_epi32(mint::r);
        const __m128i Imag = _mm_set1_epi32(imag.a);
        mint ww = one, xx = one, yy = one;
        u <<= 2;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = v + v;
            int j3 = j2 + v;
            for (; j0 < v; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
              const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
              const __m128i T0M1 = montgomery_sub_128(T0, T1, m2, m0);
              const __m128i T2M3 = montgomery_mul_128(
                  montgomery_sub_128(T2, T3, m2, m0), Imag, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P1, T2P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j2),
                               montgomery_sub_128(T0P1, T2P3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_add_128(T0M1, T2M3, m2, m0));
              _mm_storeu_si128((__m128i *)(a + j3),
                               montgomery_sub_128(T0M1, T2M3, m2, m0));
            }
          } else {
            ww = xx * xx, yy = xx * imag;
            const __m128i WW = _mm_set1_epi32(ww.a);
            const __m128i XX = _mm_set1_epi32(xx.a);
            const __m128i YY = _mm_set1_epi32(yy.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
              const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
              const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
              const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
              const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
              const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
              const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
              const __m128i T0M1 = montgomery_mul_128(
                  montgomery_sub_128(T0, T1, m2, m0), XX, r, m1);
              __m128i T2M3 = montgomery_mul_128(
                  montgomery_sub_128(T2, T3, m2, m0), YY, r, m1);
              _mm_storeu_si128((__m128i *)(a + j0),
                               montgomery_add_128(T0P1, T2P3, m2, m0));
              _mm_storeu_si128(
                  (__m128i *)(a + j2),
                  montgomery_mul_128(montgomery_sub_128(T0P1, T2P3, m2, m0), WW,
                                     r, m1));
              _mm_storeu_si128((__m128i *)(a + j1),
                               montgomery_add_128(T0M1, T2M3, m2, m0));
              _mm_storeu_si128(
                  (__m128i *)(a + j3),
                  montgomery_mul_128(montgomery_sub_128(T0M1, T2M3, m2, m0), WW,
                                     r, m1));
            }
          }
          xx *= dy[__builtin_ctz(jh += 4)];
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m1 = _mm256_set1_epi32(mod);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        const __m256i r = _mm256_set1_epi32(mint::r);
        const __m256i Imag = _mm256_set1_epi32(imag.a);
        mint ww = one, xx = one, yy = one;
        u <<= 2;
        for (int jh = 0; jh < u;) {
          if (jh == 0) {
            int j0 = 0;
            int j1 = v;
            int j2 = v + v;
            int j3 = j2 + v;
            for (; j0 < v; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
              const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
              const __m256i T0M1 = montgomery_sub_256(T0, T1, m2, m0);
              const __m256i T2M3 = montgomery_mul_256(
                  montgomery_sub_256(T2, T3, m2, m0), Imag, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P1, T2P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j2),
                                  montgomery_sub_256(T0P1, T2P3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_add_256(T0M1, T2M3, m2, m0));
              _mm256_storeu_si256((__m256i *)(a + j3),
                                  montgomery_sub_256(T0M1, T2M3, m2, m0));
            }
          } else {
            ww = xx * xx, yy = xx * imag;
            const __m256i WW = _mm256_set1_epi32(ww.a);
            const __m256i XX = _mm256_set1_epi32(xx.a);
            const __m256i YY = _mm256_set1_epi32(yy.a);
            int j0 = jh * v;
            int j1 = j0 + v;
            int j2 = j1 + v;
            int j3 = j2 + v;
            int je = j1;
            for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
              const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
              const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
              const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
              const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
              const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
              const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
              const __m256i T0M1 = montgomery_mul_256(
                  montgomery_sub_256(T0, T1, m2, m0), XX, r, m1);
              const __m256i T2M3 = montgomery_mul_256(
                  montgomery_sub_256(T2, T3, m2, m0), YY, r, m1);
              _mm256_storeu_si256((__m256i *)(a + j0),
                                  montgomery_add_256(T0P1, T2P3, m2, m0));
              _mm256_storeu_si256(
                  (__m256i *)(a + j2),
                  montgomery_mul_256(montgomery_sub_256(T0P1, T2P3, m2, m0), WW,
                                     r, m1));
              _mm256_storeu_si256((__m256i *)(a + j1),
                                  montgomery_add_256(T0M1, T2M3, m2, m0));
              _mm256_storeu_si256(
                  (__m256i *)(a + j3),
                  montgomery_mul_256(montgomery_sub_256(T0M1, T2M3, m2, m0), WW,
                                     r, m1));
            }
          }
          xx *= dy[__builtin_ctz(jh += 4)];
        }
      }
      u >>= 4;
      v <<= 2;
    }
    if (k & 1) {
      v = 1 << (k - 1);
      if (v < 8) {
        for (int j = 0; j < v; ++j) {
          mint ajv = a[j] - a[j + v];
          a[j] += a[j + v];
          a[j + v] = ajv;
        }
      } else {
        const __m256i m0 = _mm256_set1_epi32(0);
        const __m256i m2 = _mm256_set1_epi32(mod + mod);
        int j0 = 0;
        int j1 = v;
        for (; j0 < v; j0 += 8, j1 += 8) {
          const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
          const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
          __m256i naj = montgomery_add_256(T0, T1, m2, m0);
          __m256i najv = montgomery_sub_256(T0, T1, m2, m0);
          _mm256_storeu_si256((__m256i *)(a + j0), naj);
          _mm256_storeu_si256((__m256i *)(a + j1), najv);
        }
      }
    }
    if (normalize) {
      mint invn = mint(n).inverse();
      for (int i = 0; i < n; i++) a[i] *= invn;
    }
  }

  __attribute__((target("avx2"))) void inplace_multiply(
      int l1, int l2, int zero_padding = true) {
    int l = l1 + l2 - 1;
    int M = 4;
    while (M < l) M <<= 1;
    if (zero_padding) {
      for (int i = l1; i < M; i++) ntt_inner::_buf1[i] = 0;
      for (int i = l2; i < M; i++) ntt_inner::_buf2[i] = 0;
    }
    const __m256i m0 = _mm256_set1_epi32(0);
    const __m256i m1 = _mm256_set1_epi32(mod);
    const __m256i r = _mm256_set1_epi32(mint::r);
    const __m256i N2 = _mm256_set1_epi32(mint::n2);
    for (int i = 0; i < l1; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
      __m256i b = montgomery_mul_256(a, N2, r, m1);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), b);
    }
    for (int i = 0; i < l2; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
      __m256i b = montgomery_mul_256(a, N2, r, m1);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf2 + i), b);
    }
    ntt(buf1, M);
    ntt(buf2, M);
    for (int i = 0; i < M; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
      __m256i b = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
      __m256i c = montgomery_mul_256(a, b, r, m1);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), c);
    }
    intt(buf1, M, false);
    const __m256i INVM = _mm256_set1_epi32((mint(M).inverse()).a);
    for (int i = 0; i < l; i += 8) {
      __m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
      __m256i b = montgomery_mul_256(a, INVM, r, m1);
      __m256i c = my256_mulhi_epu32(my256_mullo_epu32(b, r), m1);
      __m256i d = _mm256_and_si256(_mm256_cmpgt_epi32(c, m0), m1);
      __m256i e = _mm256_sub_epi32(d, c);
      _mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), e);
    }
  }

  void ntt(vector<mint> &a) {
    int M = (int)a.size();
    for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
    ntt(buf1, M);
    for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
  }

  void intt(vector<mint> &a) {
    int M = (int)a.size();
    for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
    intt(buf1, M, true);
    for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
  }

  vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
    if (a.size() == 0 && b.size() == 0) return vector<mint>{};
    int l = a.size() + b.size() - 1;
    if (min<int>(a.size(), b.size()) <= 40) {
      vector<mint> s(l);
      for (int i = 0; i < (int)a.size(); ++i)
        for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
      return s;
    }
    assert(l <= ntt_inner::SZ_FFT_BUF);
    int M = 4;
    while (M < l) M <<= 1;
    for (int i = 0; i < (int)a.size(); ++i) buf1[i].a = a[i].a;
    for (int i = (int)a.size(); i < M; ++i) buf1[i].a = 0;
    for (int i = 0; i < (int)b.size(); ++i) buf2[i].a = b[i].a;
    for (int i = (int)b.size(); i < M; ++i) buf2[i].a = 0;
    ntt(buf1, M);
    ntt(buf2, M);
    for (int i = 0; i < M; ++i)
      buf1[i].a = mint::reduce(uint64_t(buf1[i].a) * buf2[i].a);
    intt(buf1, M, false);
    vector<mint> s(l);
    mint invm = mint(M).inverse();
    for (int i = 0; i < l; ++i) s[i] = buf1[i] * invm;
    return s;
  }

  void ntt_doubling(vector<mint> &a) {
    int M = (int)a.size();
    for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
    intt(buf1, M);
    mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
    for (int i = 0; i < M; i++) buf1[i] *= r, r *= zeta;
    ntt(buf1, M);
    a.resize(2 * M);
    for (int i = 0; i < M; i++) a[M + i].a = buf1[i].a;
  }
};


namespace ArbitraryNTT {
using i64 = int64_t;
using u128 = __uint128_t;
constexpr int32_t m0 = 167772161;
constexpr int32_t m1 = 469762049;
constexpr int32_t m2 = 754974721;
using mint0 = LazyMontgomeryModInt<m0>;
using mint1 = LazyMontgomeryModInt<m1>;
using mint2 = LazyMontgomeryModInt<m2>;
constexpr int r01 = mint1(m0).inverse().get();
constexpr int r02 = mint2(m0).inverse().get();
constexpr int r12 = mint2(m1).inverse().get();
constexpr int r02r12 = i64(r02) * r12 % m2;
constexpr i64 w1 = m0;
constexpr i64 w2 = i64(m0) * m1;

template <typename T, typename submint>
vector<submint> mul(const vector<T> &a, const vector<T> &b) {
  static NTT<submint> ntt;
  vector<submint> s(a.size()), t(b.size());
  for (int i = 0; i < (int)a.size(); ++i) s[i] = i64(a[i] % submint::get_mod());
  for (int i = 0; i < (int)b.size(); ++i) t[i] = i64(b[i] % submint::get_mod());
  return ntt.multiply(s, t);
}

template <typename T>
vector<int> multiply(const vector<T> &s, const vector<T> &t, int mod) {
  auto d0 = mul<T, mint0>(s, t);
  auto d1 = mul<T, mint1>(s, t);
  auto d2 = mul<T, mint2>(s, t);
  int n = d0.size();
  vector<int> ret(n);
  const int W1 = w1 % mod;
  const int W2 = w2 % mod;
  for (int i = 0; i < n; i++) {
    int n1 = d1[i].get(), n2 = d2[i].get(), a = d0[i].get();
    int b = i64(n1 + m1 - a) * r01 % m1;
    int c = (i64(n2 + m2 - a) * r02r12 + i64(m2 - b) * r12) % m2;
    ret[i] = (i64(a) + i64(b) * W1 + i64(c) * W2) % mod;
  }
  return ret;
}

template <typename mint>
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
  if (a.size() == 0 && b.size() == 0) return {};
  if (min<int>(a.size(), b.size()) < 128) {
    vector<mint> ret(a.size() + b.size() - 1);
    for (int i = 0; i < (int)a.size(); ++i)
      for (int j = 0; j < (int)b.size(); ++j) ret[i + j] += a[i] * b[j];
    return ret;
  }
  vector<int> s(a.size()), t(b.size());
  for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i].get();
  for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i].get();
  vector<int> u = multiply<int>(s, t, mint::get_mod());
  vector<mint> ret(u.size());
  for (int i = 0; i < (int)u.size(); ++i) ret[i] = mint(u[i]);
  return ret;
}

template <typename T>
vector<u128> multiply_u128(const vector<T> &s, const vector<T> &t) {
  if (s.size() == 0 && t.size() == 0) return {};
  if (min<int>(s.size(), t.size()) < 128) {
    vector<u128> ret(s.size() + t.size() - 1);
    for (int i = 0; i < (int)s.size(); ++i)
      for (int j = 0; j < (int)t.size(); ++j) ret[i + j] += i64(s[i]) * t[j];
    return ret;
  }
  auto d0 = mul<T, mint0>(s, t);
  auto d1 = mul<T, mint1>(s, t);
  auto d2 = mul<T, mint2>(s, t);
  int n = d0.size();
  vector<u128> ret(n);
  for (int i = 0; i < n; i++) {
    i64 n1 = d1[i].get(), n2 = d2[i].get();
    i64 a = d0[i].get();
    i64 b = (n1 + m1 - a) * r01 % m1;
    i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2;
    ret[i] = a + b * w1 + u128(c) * w2;
  }
  return ret;
}
}  // namespace ArbitraryNTT


namespace MultiPrecisionIntegerImpl {
struct TENS {
  static constexpr int offset = 30;
  constexpr TENS() : _tend() {
    _tend[offset] = 1;
    for (int i = 1; i <= offset; i++) {
      _tend[offset + i] = _tend[offset + i - 1] * 10.0;
      _tend[offset - i] = 1.0 / _tend[offset + i];
    }
  }
  long double ten_ld(int n) const {
    assert(-offset <= n and n <= offset);
    return _tend[n + offset];
  }

 private:
  long double _tend[offset * 2 + 1];
};
}  // namespace MultiPrecisionIntegerImpl

// 0 は neg=false, dat={} として扱う
struct MultiPrecisionInteger {
  using M = MultiPrecisionInteger;
  inline constexpr static MultiPrecisionIntegerImpl::TENS tens = {};

  static constexpr int D = 1000000000;
  static constexpr int logD = 9;
  bool neg;
  vector<int> dat;

  MultiPrecisionInteger() : neg(false), dat() {}

  MultiPrecisionInteger(bool n, const vector<int>& d) : neg(n), dat(d) {}

  template <typename I,
            enable_if_t<internal::is_broadly_integral_v<I>>* = nullptr>
  MultiPrecisionInteger(I x) : neg(false) {
    if constexpr (internal::is_broadly_signed_v<I>) {
      if (x < 0) neg = true, x = -x;
    }
    while (x) dat.push_back(x % D), x /= D;
  }

  MultiPrecisionInteger(const string& S) : neg(false) {
    assert(!S.empty());
    if (S.size() == 1u && S[0] == '0') return;
    int l = 0;
    if (S[0] == '-') ++l, neg = true;
    for (int ie = S.size(); l < ie; ie -= logD) {
      int is = max(l, ie - logD);
      long long x = 0;
      for (int i = is; i < ie; i++) x = x * 10 + S[i] - '0';
      dat.push_back(x);
    }
    while(!dat.empty() and dat.back() == 0) dat.pop_back();
  }

  friend M operator+(const M& lhs, const M& rhs) {
    if (lhs.neg == rhs.neg) return {lhs.neg, _add(lhs.dat, rhs.dat)};
    if (_leq(lhs.dat, rhs.dat)) {
      // |l| <= |r|
      auto c = _sub(rhs.dat, lhs.dat);
      bool n = _is_zero(c) ? false : rhs.neg;
      return {n, c};
    }
    auto c = _sub(lhs.dat, rhs.dat);
    bool n = _is_zero(c) ? false : lhs.neg;
    return {n, c};
  }
  friend M operator-(const M& lhs, const M& rhs) { return lhs + (-rhs); }

  friend M operator*(const M& lhs, const M& rhs) {
    auto c = _mul(lhs.dat, rhs.dat);
    bool n = _is_zero(c) ? false : (lhs.neg ^ rhs.neg);
    return {n, c};
  }
  friend pair<M, M> divmod(const M& lhs, const M& rhs) {
    auto dm = _divmod_newton(lhs.dat, rhs.dat);
    bool dn = _is_zero(dm.first) ? false : lhs.neg != rhs.neg;
    bool mn = _is_zero(dm.second) ? false : lhs.neg;
    return {M{dn, dm.first}, M{mn, dm.second}};
  }
  friend M operator/(const M& lhs, const M& rhs) {
    return divmod(lhs, rhs).first;
  }
  friend M operator%(const M& lhs, const M& rhs) {
    return divmod(lhs, rhs).second;
  }

  M& operator+=(const M& rhs) { return (*this) = (*this) + rhs; }
  M& operator-=(const M& rhs) { return (*this) = (*this) - rhs; }
  M& operator*=(const M& rhs) { return (*this) = (*this) * rhs; }
  M& operator/=(const M& rhs) { return (*this) = (*this) / rhs; }
  M& operator%=(const M& rhs) { return (*this) = (*this) % rhs; }

  M operator-() const {
    if (is_zero()) return *this;
    return {!neg, dat};
  }
  M operator+() const { return *this; }
  friend M abs(const M& m) { return {false, m.dat}; }
  bool is_zero() const { return _is_zero(dat); }

  friend bool operator==(const M& lhs, const M& rhs) {
    return lhs.neg == rhs.neg && lhs.dat == rhs.dat;
  }
  friend bool operator!=(const M& lhs, const M& rhs) {
    return lhs.neg != rhs.neg || lhs.dat != rhs.dat;
  }
  friend bool operator<(const M& lhs, const M& rhs) {
    if (lhs == rhs) return false;
    return _neq_lt(lhs, rhs);
  }
  friend bool operator<=(const M& lhs, const M& rhs) {
    if (lhs == rhs) return true;
    return _neq_lt(lhs, rhs);
  }
  friend bool operator>(const M& lhs, const M& rhs) {
    if (lhs == rhs) return false;
    return _neq_lt(rhs, lhs);
  }
  friend bool operator>=(const M& lhs, const M& rhs) {
    if (lhs == rhs) return true;
    return _neq_lt(rhs, lhs);
  }

  // a * 10^b (1 <= |a| < 10) の形で渡す
  // 相対誤差：10^{-16} ~ 10^{-19} 程度 (処理系依存)
  pair<long double, int> dfp() const {
    if (is_zero()) return {0, 0};
    int l = max<int>(0, _size() - 3);
    int b = logD * l;
    string prefix{};
    for (int i = _size() - 1; i >= l; i--) {
      prefix += _itos(dat[i], i != _size() - 1);
    }
    b += prefix.size() - 1;
    long double a = 0;
    for (auto& c : prefix) a = a * 10.0 + (c - '0');
    a *= tens.ten_ld(-((int)prefix.size()) + 1);
    a = clamp<long double>(a, 1.0, nextafterl(10.0, 1.0));
    if (neg) a = -a;
    return {a, b};
  }
  string to_string() const {
    if (is_zero()) return "0";
    string res;
    if (neg) res.push_back('-');
    for (int i = _size() - 1; i >= 0; i--) {
      res += _itos(dat[i], i != _size() - 1);
    }
    return res;
  }
  long double to_ld() const {
    auto [a, b] = dfp();
    if (-tens.offset <= b and b <= tens.offset) {
      return a * tens.ten_ld(b);
    }
    return a * powl(10, b);
  }
  long long to_ll() const {
    long long res = _to_ll(dat);
    return neg ? -res : res;
  }
  __int128_t to_i128() const {
    __int128_t res = _to_i128(dat);
    return neg ? -res : res;
  }

  friend istream& operator>>(istream& is, M& m) {
    string s;
    is >> s;
    m = M{s};
    return is;
  }

  friend ostream& operator<<(ostream& os, const M& m) {
    return os << m.to_string();
  }

  // 内部の関数をテスト
  static void _test_private_function(const M&, const M&);

 private:
  // size
  int _size() const { return dat.size(); }
  // a == b
  static bool _eq(const vector<int>& a, const vector<int>& b) { return a == b; }
  // a < b
  static bool _lt(const vector<int>& a, const vector<int>& b) {
    if (a.size() != b.size()) return a.size() < b.size();
    for (int i = a.size() - 1; i >= 0; i--) {
      if (a[i] != b[i]) return a[i] < b[i];
    }
    return false;
  }
  // a <= b
  static bool _leq(const vector<int>& a, const vector<int>& b) {
    return _eq(a, b) || _lt(a, b);
  }
  // a < b (s.t. a != b)
  static bool _neq_lt(const M& lhs, const M& rhs) {
    assert(lhs != rhs);
    if (lhs.neg != rhs.neg) return lhs.neg;
    bool f = _lt(lhs.dat, rhs.dat);
    if (f) return !lhs.neg;
    return lhs.neg;
  }
  // a == 0
  static bool _is_zero(const vector<int>& a) { return a.empty(); }
  // a == 1
  static bool _is_one(const vector<int>& a) {
    return (int)a.size() == 1 && a[0] == 1;
  }
  // 末尾 0 を削除
  static void _shrink(vector<int>& a) {
    while (a.size() && a.back() == 0) a.pop_back();
  }
  // 末尾 0 を削除
  void _shrink() {
    while (_size() && dat.back() == 0) dat.pop_back();
  }
  // a + b
  static vector<int> _add(const vector<int>& a, const vector<int>& b) {
    vector<int> c(max(a.size(), b.size()) + 1);
    for (int i = 0; i < (int)a.size(); i++) c[i] += a[i];
    for (int i = 0; i < (int)b.size(); i++) c[i] += b[i];
    for (int i = 0; i < (int)c.size() - 1; i++) {
      if (c[i] >= D) c[i] -= D, c[i + 1]++;
    }
    _shrink(c);
    return c;
  }
  // a - b
  static vector<int> _sub(const vector<int>& a, const vector<int>& b) {
    assert(_leq(b, a));
    vector<int> c{a};
    int borrow = 0;
    for (int i = 0; i < (int)a.size(); i++) {
      if (i < (int)b.size()) borrow += b[i];
      c[i] -= borrow;
      borrow = 0;
      if (c[i] < 0) c[i] += D, borrow = 1;
    }
    assert(borrow == 0);
    _shrink(c);
    return c;
  }
  // a * b (fft)
  static vector<int> _mul_fft(const vector<int>& a, const vector<int>& b) {
    if (a.empty() || b.empty()) return {};
    auto m = ArbitraryNTT::multiply_u128(a, b);
    vector<int> c;
    c.reserve(m.size() + 3);
    __uint128_t x = 0;
    for (int i = 0;; i++) {
      if (i >= (int)m.size() && x == 0) break;
      if (i < (int)m.size()) x += m[i];
      c.push_back(x % D);
      x /= D;
    }
    _shrink(c);
    return c;
  }
  // a * b (naive)
  static vector<int> _mul_naive(const vector<int>& a, const vector<int>& b) {
    if (a.empty() || b.empty()) return {};
    vector<long long> prod(a.size() + b.size() - 1 + 1);
    for (int i = 0; i < (int)a.size(); i++) {
      for (int j = 0; j < (int)b.size(); j++) {
        long long p = 1LL * a[i] * b[j];
        prod[i + j] += p;
        if (prod[i + j] >= (4LL * D * D)) {
          prod[i + j] -= 4LL * D * D;
          prod[i + j + 1] += 4LL * D;
        }
      }
    }
    vector<int> c(prod.size() + 1);
    long long x = 0;
    int i = 0;
    for (; i < (int)prod.size(); i++) x += prod[i], c[i] = x % D, x /= D;
    while (x) c[i] = x % D, x /= D, i++;
    _shrink(c);
    return c;
  }
  // a * b
  static vector<int> _mul(const vector<int>& a, const vector<int>& b) {
    if (_is_zero(a) || _is_zero(b)) return {};
    if (_is_one(a)) return b;
    if (_is_one(b)) return a;
    if (min<int>(a.size(), b.size()) <= 128) {
      return a.size() < b.size() ? _mul_naive(b, a) : _mul_naive(a, b);
    }
    return _mul_fft(a, b);
  }
  // 0 <= A < 1e18, 1 <= B < 1e9
  static pair<vector<int>, vector<int>> _divmod_li(const vector<int>& a,
                                                   const vector<int>& b) {
    assert(0 <= (int)a.size() && (int)a.size() <= 2);
    assert((int)b.size() == 1);
    long long va = _to_ll(a);
    int vb = b[0];
    return {_integer_to_vec(va / vb), _integer_to_vec(va % vb)};
  }
  // 0 <= A < 1e18, 1 <= B < 1e18
  static pair<vector<int>, vector<int>> _divmod_ll(const vector<int>& a,
                                                   const vector<int>& b) {
    assert(0 <= (int)a.size() && (int)a.size() <= 2);
    assert(1 <= (int)b.size() && (int)b.size() <= 2);
    long long va = _to_ll(a), vb = _to_ll(b);
    return {_integer_to_vec(va / vb), _integer_to_vec(va % vb)};
  }
  // 1 <= B < 1e9
  static pair<vector<int>, vector<int>> _divmod_1e9(const vector<int>& a,
                                                    const vector<int>& b) {
    assert((int)b.size() == 1);
    if (b[0] == 1) return {a, {}};
    if ((int)a.size() <= 2) return _divmod_li(a, b);
    vector<int> quo(a.size());
    long long d = 0;
    int b0 = b[0];
    for (int i = a.size() - 1; i >= 0; i--) {
      d = d * D + a[i];
      assert(d < 1LL * D * b0);
      int q = d / b0, r = d % b0;
      quo[i] = q, d = r;
    }
    _shrink(quo);
    return {quo, d ? vector<int>{int(d)} : vector<int>{}};
  }
  // 0 <= A, 1 <= B
  static pair<vector<int>, vector<int>> _divmod_naive(const vector<int>& a,
                                                      const vector<int>& b) {
    if (_is_zero(b)) {
      cerr << "Divide by Zero Exception" << endl;
      exit(1);
    }
    assert(1 <= (int)b.size());
    if ((int)b.size() == 1) return _divmod_1e9(a, b);
    if (max<int>(a.size(), b.size()) <= 2) return _divmod_ll(a, b);
    if (_lt(a, b)) return {{}, a};
    // B >= 1e9, A >= B
    int norm = D / (b.back() + 1);
    vector<int> x = _mul(a, {norm});
    vector<int> y = _mul(b, {norm});
    int yb = y.back();
    vector<int> quo(x.size() - y.size() + 1);
    vector<int> rem(x.end() - y.size(), x.end());
    for (int i = quo.size() - 1; i >= 0; i--) {
      if (rem.size() < y.size()) {
        // do nothing
      } else if (rem.size() == y.size()) {
        if (_leq(y, rem)) {
          quo[i] = 1, rem = _sub(rem, y);
        }
      } else {
        assert(y.size() + 1 == rem.size());
        long long rb = 1LL * rem[rem.size() - 1] * D + rem[rem.size() - 2];
        int q = rb / yb;
        vector<int> yq = _mul(y, {q});
        // 真の商は q-2 以上 q+1 以下だが自信が無いので念のため while を回す
        while (_lt(rem, yq)) q--, yq = _sub(yq, y);
        rem = _sub(rem, yq);
        while (_leq(y, rem)) q++, rem = _sub(rem, y);
        quo[i] = q;
      }
      if (i) rem.insert(begin(rem), x[i - 1]);
    }
    _shrink(quo), _shrink(rem);
    auto [q2, r2] = _divmod_1e9(rem, {norm});
    assert(_is_zero(r2));
    return {quo, q2};
  }

  // 0 <= A, 1 <= B
  static pair<vector<int>, vector<int>> _divmod_dc(const vector<int>& a,
                                                   const vector<int>& b);

  // 1 / a を 絶対誤差 B^{-deg} で求める
  static vector<int> _calc_inv(const vector<int>& a, int deg) {
    assert(!a.empty() && D / 2 <= a.back() and a.back() < D);
    int k = deg, c = a.size();
    while (k > 64) k = (k + 1) / 2;
    vector<int> z(c + k + 1);
    z.back() = 1;
    z = _divmod_naive(z, a).first;
    while (k < deg) {
      vector<int> s = _mul(z, z);
      s.insert(begin(s), 0);
      int d = min(c, 2 * k + 1);
      vector<int> t{end(a) - d, end(a)}, u = _mul(s, t);
      u.erase(begin(u), begin(u) + d);
      vector<int> w(k + 1), w2 = _add(z, z);
      copy(begin(w2), end(w2), back_inserter(w));
      z = _sub(w, u);
      z.erase(begin(z));
      k *= 2;
    }
    z.erase(begin(z), begin(z) + k - deg);
    return z;
  }

  static pair<vector<int>, vector<int>> _divmod_newton(const vector<int>& a,
                                                       const vector<int>& b) {
    if (_is_zero(b)) {
      cerr << "Divide by Zero Exception" << endl;
      exit(1);
    }
    if ((int)b.size() <= 64) return _divmod_naive(a, b);
    if ((int)a.size() - (int)b.size() <= 64) return _divmod_naive(a, b);
    int norm = D / (b.back() + 1);
    vector<int> x = _mul(a, {norm});
    vector<int> y = _mul(b, {norm});
    int s = x.size(), t = y.size();
    int deg = s - t + 2;
    vector<int> z = _calc_inv(y, deg);
    vector<int> q = _mul(x, z);
    q.erase(begin(q), begin(q) + t + deg);
    vector<int> yq = _mul(y, {q});
    while (_lt(x, yq)) q = _sub(q, {1}), yq = _sub(yq, y);
    vector<int> r = _sub(x, yq);
    while (_leq(y, r)) q = _add(q, {1}), r = _sub(r, y);
    _shrink(q), _shrink(r);
    auto [q2, r2] = _divmod_1e9(r, {norm});
    assert(_is_zero(r2));
    return {q, q2};
  }

  // int -> string
  // 先頭かどうかに応じて zero padding するかを決める
  static string _itos(int x, bool zero_padding) {
    assert(0 <= x && x < D);
    string res;
    for (int i = 0; i < logD; i++) {
      res.push_back('0' + x % 10), x /= 10;
    }
    if (!zero_padding) {
      while (res.size() && res.back() == '0') res.pop_back();
      assert(!res.empty());
    }
    reverse(begin(res), end(res));
    return res;
  }

  // convert ll to vec
  template <typename I,
            enable_if_t<internal::is_broadly_integral_v<I>>* = nullptr>
  static vector<int> _integer_to_vec(I x) {
    if constexpr (internal::is_broadly_signed_v<I>) {
      assert(x >= 0);
    }
    vector<int> res;
    while (x) res.push_back(x % D), x /= D;
    return res;
  }

  static long long _to_ll(const vector<int>& a) {
    long long res = 0;
    for (int i = (int)a.size() - 1; i >= 0; i--) res = res * D + a[i];
    return res;
  }

  static __int128_t _to_i128(const vector<int>& a) {
    __int128_t res = 0;
    for (int i = (int)a.size() - 1; i >= 0; i--) res = res * D + a[i];
    return res;
  }

  static void _dump(const vector<int>& a, string s = "") {
    if (!s.empty()) cerr << s << " : ";
    cerr << "{ ";
    for (int i = 0; i < (int)a.size(); i++) cerr << a[i] << ", ";
    cerr << "}" << endl;
  }
};

using bigint = MultiPrecisionInteger;

/**
 * @brief 多倍長整数
 */


//




namespace internal {


using namespace std;

// a mod p
template <typename T>
T safe_mod(T a, T p) {
  a %= p;
  if constexpr (is_broadly_signed_v<T>) {
    if (a < 0) a += p;
  }
  return a;
}

// 返り値：pair(g, x)
// s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
template <typename T>
pair<T, T> inv_gcd(T a, T p) {
  static_assert(is_broadly_signed_v<T>);
  a = safe_mod(a, p);
  if (a == 0) return {p, 0};
  T b = p, x = 1, y = 0;
  while (a != 0) {
    T q = b / a;
    swap(a, b %= a);
    swap(x, y -= q * x);
  }
  if (y < 0) y += p / b;
  return {b, y};
}

// 返り値 : a^{-1} mod p
// gcd(a, p) != 1 が必要
template <typename T>
T inv(T a, T p) {
  static_assert(is_broadly_signed_v<T>);
  a = safe_mod(a, p);
  T b = p, x = 1, y = 0;
  while (a != 0) {
    T q = b / a;
    swap(a, b %= a);
    swap(x, y -= q * x);
  }
  assert(b == 1);
  return y < 0 ? y + p : y;
}

// T : 底の型
// U : T*T がオーバーフローしない かつ 指数の型
template <typename T, typename U>
T modpow(T a, U n, T p) {
  a = safe_mod(a, p);
  T ret = 1 % p;
  while (n != 0) {
    if (n % 2 == 1) ret = U(ret) * a % p;
    a = U(a) * a % p;
    n /= 2;
  }
  return ret;
}

// 返り値 : pair(rem, mod)
// 解なしのときは {0, 0} を返す
template <typename T>
pair<T, T> crt(const vector<T>& r, const vector<T>& m) {
  static_assert(is_broadly_signed_v<T>);
  assert(r.size() == m.size());
  int n = int(r.size());
  T r0 = 0, m0 = 1;
  for (int i = 0; i < n; i++) {
    assert(1 <= m[i]);
    T r1 = safe_mod(r[i], m[i]), m1 = m[i];
    if (m0 < m1) swap(r0, r1), swap(m0, m1);
    if (m0 % m1 == 0) {
      if (r0 % m1 != r1) return {0, 0};
      continue;
    }
    auto [g, im] = inv_gcd(m0, m1);
    T u1 = m1 / g;
    if ((r1 - r0) % g) return {0, 0};
    T x = (r1 - r0) / g % u1 * im % u1;
    r0 += x * m0;
    m0 *= u1;
    if (r0 < 0) r0 += m0;
  }
  return {r0, m0};
}

}  // namespace internal





using namespace std;

struct Timer {
  chrono::high_resolution_clock::time_point st;

  Timer() { reset(); }
  void reset() { st = chrono::high_resolution_clock::now(); }

  long long elapsed() {
    auto ed = chrono::high_resolution_clock::now();
    return chrono::duration_cast<chrono::milliseconds>(ed - st).count();
  }
  long long operator()() { return elapsed(); }
};




// Prime Sieve {2, 3, 5, 7, 11, 13, 17, ...}
vector<int> prime_enumerate(int N) {
  vector<bool> sieve(N / 3 + 1, 1);
  for (int p = 5, d = 4, i = 1, sqn = sqrt(N); p <= sqn; p += d = 6 - d, i++) {
    if (!sieve[i]) continue;
    for (int q = p * p / 3, r = d * p / 3 + (d * p % 3 == 2), s = 2 * p,
             qe = sieve.size();
         q < qe; q += r = s - r)
      sieve[q] = 0;
  }
  vector<int> ret{2, 3};
  for (int p = 5, d = 4, i = 1; p <= N; p += d = 6 - d, i++)
    if (sieve[i]) ret.push_back(p);
  while (!ret.empty() && ret.back() > N) ret.pop_back();
  return ret;
}


//
using namespace Nyaan;

void Nyaan::solve() {
  Timer timer;
  auto ps = prime_enumerate(50010);

  int T = 1;
  in(T);
  vi Ns(T);
  in(Ns);
  set<int> Nset;
  each(N, Ns) Nset.insert(N);
  map<int, string> ans;

  vi sobeki(50001, -1);
  rep(i, sz(ps)) {
    int p = ps[i];
    if (p > 50000) break;
    ll x = p;
    while (true) {
      if (x > 50000) break;
      sobeki[x] = i;
      x *= p;
    }
  }
  vi nums;
  vi prods;

  for (int n = 1; n <= 50000; n++) {
    if (sobeki[n] != -1) {
      int p = ps[sobeki[n]];
      if (p == n) {
        int prod = 1;
        each(num, nums) prod = 1LL * prod * num % n;
        nums.push_back(n);
        prods.push_back(prod);
      } else {
        int s = sobeki[n];
        nums[s] *= p;
        prods[s] = 1;
        rep(i, sz(nums)) {
          if (i != s) prods[s] = 1LL * prods[s] * nums[i] % nums[s];
        }
      }
      rep(i, sz(nums)) {
        if (i != sobeki[n]) prods[i] = 1LL * prods[i] * p % nums[i];
      }
    }
    if (n <= 10) {
      trc(nums);
      trc(prods);
    }

    if (Nset.count(n)) {
      if (n == 1) {
        ans[n] = "1/1";
      } else if (n == 2) {
        ans[n] = "1/2";
      } else {
        vi dens;
        rep(i, sz(prods)) {
          int inv = internal::inv(prods[i], nums[i]);
          dens.push_back(inv);
        };
        double dd = 0;
        rep(i, sz(nums)) dd += 1.0 * dens[i] / nums[i];
        ll d = llround(dd);
        string bns;
        rep(i, sz(nums)) {
          ll m = nums[i];
          if (i < d) {
            bns += "-" + to_string(m - dens[i]) + "/" + to_string(m);
          } else {
            if (i) bns += "+";
            bns += to_string(dens[i]) + "/" + to_string(m);
          }
        }
        ans[n] = bns;
      }
    }
  }

  each(N, Ns) {
    assert(ans.count(N));
    out(ans[N]);
  }
  trc2(timer.elapsed());
}