#define INF 4'000'000'000'000'000'037LL

// https://github.com/miscalculation53/library/tree/wip/template/template_all.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_all_but_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_types.hpp

#include <bits/stdc++.h>
using namespace std;

using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using tllll = tuple<ll, ll, ll, ll>;

#define vc vector
template <class T>
using vvc = vc<vc<T>>;

using vl = vc<ll>;
using vstr = vc<string>;

template <class T>
using pql = priority_queue<T, vc<T>, greater<T>>;

using i128 = __int128_t;
using u128 = __uint128_t;
i128 stoi128(const string &s)
{
  const bool neg = s.front() == '-';
  u128 res = 0;
  for (int i = neg; i < (int)s.size(); i++)
    res = 10 * res + s[i] - '0';
  if (neg)
    return -i128(res - 1) - 1;
  return i128(res);
}
string i128tos(i128 x)
{
  if (x == 0) return "0";
  string sign = "", res = "";
  u128 ux;
  if (x < 0)
    ux = u128(-(x + 1)) + 1, sign = "-";
  else
    ux = x;
  while (ux > 0)
  {
    res += '0' + ux % 10;
    ux /= 10;
  }
  reverse(res.begin(), res.end());
  return sign + res;
}
istream &operator>>(istream &is, i128 &a)
{
  string s;
  is >> s;
  a = stoi128(s);
  return is;
}
ostream &operator<<(ostream &os, const i128 &a)
{
  os << i128tos(a);
  return os;
}

#define cauto const auto
// https://github.com/miscalculation53/library/tree/wip/template/template_rep.hpp

#define overload4(_1,_2,_3,_4,name,...) name
#define rep2(i,l,r) for (ll i = ll(l), rrrrr = ll(r); i < rrrrr; i++)
#define rep(...) overload4(__VA_ARGS__, rep3, rep2, rep1)(__VA_ARGS__)
#define repi1(i,n) for (int i = 0, nnnnn = int(n); i < nnnnn; i++)
#define repi2(i,l,r) for (int i = int(l), rrrrr = int(r); i < rrrrr; i++)
#define repi3(i,l,r,d) for (int i = int(l), rrrrr = int(r), ddddd = int(d); ddddd > 0 ? i < rrrrr : i > rrrrr; i += d)
#define repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)

#define fe(...) for (auto __VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
#define fem(...) for (auto &__VA_ARGS__)
// https://github.com/miscalculation53/library/tree/wip/template/template_math.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/is_integral_ext.hpp

template <class T>
constexpr bool is_integral_ext = is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>;

template <class T>
constexpr bool is_signed_ext = is_signed_v<T> || is_same_v<T, i128>;

template <class T>
constexpr bool is_unsigned_ext = is_unsigned_v<T> || is_same_v<T, u128>;

template <class T, class U>
inline bool chmin(T &a, U b) { return a > b ? a = b, true : false; }
template <class T, class U>
inline bool chmax(T &a, U b) { return a < b ? a = b, true : false; }

template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T divfloor(U a, V b) { return T(a) / T(b) - (T(a) % T(b) && (T(a) ^ T(b)) < 0); }
template <class T = ll, class U, class V, typename = enable_if_t<is_integral_ext<U> && is_integral_ext<V>>>
inline constexpr T safemod(U a, V b) { return T(a) - T(b) * divfloor<T>(a, b); }

template <class T = ll, class U, class V>
constexpr T ipow(U a, V b)
{
  assert(b >= 0);
  if (b == 0)
    return 1;
  if (a == 0 || a == 1)
    return a;
  if (a < 0 && a == -1)
    return b & 1 ? -1 : 1;

  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
      res *= tmp;
    b >>= 1;
    if (b == 0)
      break;
    tmp *= tmp;
  }
  return res;
}

// https://github.com/miscalculation53/library/tree/wip/template/template_vector.hpp

#define ALL(a) (a).begin(), (a).end()
template <class T = ll, class V>
inline T SZ(const V &x) { return x.size(); }
#define eb emplace_back

#define LMD(x,fx) ([&](auto x) { return fx; })

// https://github.com/miscalculation53/library/tree/wip/template/template_algo.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/resolved_value.hpp

template <class T, auto x, enable_if_t<is_invocable_v<decltype(x)>, long> = 0>
const T &resolved_value()
{
  static const T value = T(x());
  return value;
}

template <class T>
vvc<T> top(const vvc<T> &a)
{
  if (a.empty())
    return {};
  const int n = a.size(), m = a[0].size();
  vvc<T> b(m, vc<T>(n));
  repi(i, n)
  {
    assert(SZ<int>(a[i]) == m);
    repi(j, m) b[j][i] = a[i][j];
  }
  return b;
}
vstr top(const vstr &a)
{
  vvc<char> a_(a.size());
  repi(i, SZ<int>(a)) a_[i] = {ALL(a[i])};
  vvc<char> b_ = top(a_);
  vstr b(b_.size());
  repi(i, SZ<int>(b)) b[i] = {ALL(b_[i])};
  return b;
}

template <class T, class = void>
struct has_e0 : false_type {};
template <class T>
struct has_e0<T, void_t<decltype(T::e0())>> : true_type {};
template <class T>
inline constexpr bool has_e0_v = has_e0<T>::value;

template <class T>
struct MonoidAdd
{
  using S = T;
  static constexpr S op(S a, S b) { return a + b; }
  static constexpr S e()
  {
    if constexpr (has_e0_v<S>)
      return S::e0();
    else
      return {};
  }
};

namespace internal
{
  template <class M, class I, class = void>
  struct HasMonoidPow : false_type
  {
  };
  template <class M, class I>
  struct HasMonoidPow<M, I, void_t<decltype(M::pow(declval<const typename M::S &>(), declval<I>()))>> : true_type
  {
  };
}

template <class M>
vc<typename M::S> cuml(const vc<typename M::S> &v, int left_index = 0)
{
  const int n = v.size();
  vc<typename M::S> res(n + 1);
  res[0] = M::e();
  repi(i, n) res[i + 1] = M::op(res[i], v[i]);
  res.erase(res.begin(), res.begin() + left_index);
  return res;
}

template <class T>
vc<T> cumlsum(const vc<T> &v, int left_index = 0)
{ return cuml<MonoidAdd<T>>(v, left_index); }

constexpr array<pll, 4> DRULgrid = {{{1, 0}, {0, 1}, {-1, 0}, {0, -1}}};
constexpr array<pll, 4> DRULplane = {{{0, -1}, {1, 0}, {0, 1}, {-1, 0}}};
// https://github.com/miscalculation53/library/tree/wip/template/template_binsearch.hpp

template <class T>
struct is_random_access_iterator
{
  static constexpr bool value = is_same_v<
    typename iterator_traits<T>::iterator_category,
    random_access_iterator_tag
  >;
};
template <class T>
constexpr bool is_random_access_iterator_v = is_random_access_iterator<T>::value;

namespace internal
{
template <class T>
bool binsearch_adjacent(T a, T b)
{
  if (a < b)
    return a + 1 == b;
  if (b < a)
    return b + 1 == a;
  return false;
}
};

template <class T = ll, class Judge, class InitOk, class InitNg>
pair<T, T> binsearch(const Judge &judge, InitOk init_ok, InitNg init_ng, bool check_ok = true, bool check_ng = true)
{
  T ok(init_ok), ng(init_ng);
  if (check_ok)
    assert(judge(ok));
  if (check_ng)
    assert(!judge(ng));
  while (!internal::binsearch_adjacent(ok, ng))
  {
    T mid = (ok & ng) + ((ok ^ ng) >> 1);
    (judge(mid) ? ok : ng) = mid;
  }
  return {ok, ng};
}

// https://github.com/miscalculation53/library/tree/wip/template/template_bit.hpp

template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }

inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }

inline constexpr ll bit_ceil(ll x) { return std::bit_ceil((ull)x); }
inline constexpr ll countr_zero(ll x) { assert(x != 0); return std::countr_zero((ull)x); }

// https://github.com/miscalculation53/library/tree/wip/template/template_inout.hpp

// https://github.com/miscalculation53/library/tree/wip/template/template_dump.hpp

#define dump(...) 
#define oj(...) __VA_ARGS__

namespace fastio {
template <class T>
struct unsigned_integer
{
  using type = make_unsigned_t<T>;
};
template <>
struct unsigned_integer<i128>
{
  using type = u128;
};
template <>
struct unsigned_integer<u128>
{
  using type = u128;
};
template <class T>
using unsigned_integer_t = typename unsigned_integer<T>::type;

static constexpr uint32_t SIZ = 1 << 17;
char ibuf[SIZ];
char obuf[SIZ];
char out[100];

uint32_t pil = 0, pir = 0, por = 0;

struct Pre {
  char num[10000][4];
  constexpr Pre() : num() {
    for (int i = 0; i < 10000; i++) {
      int n = i;
      for (int j = 3; j >= 0; j--) {
        num[i][j] = n % 10 | '0';
        n /= 10;
      }
    }
  }
} constexpr pre;

inline void load() {
  memcpy(ibuf, ibuf + pil, pir - pil);
  pir = pir - pil + fread(ibuf + pir - pil, 1, SIZ - pir + pil, stdin);
  pil = 0;
  if (pir < SIZ) ibuf[pir++] = '\n';
}

inline void flush() {
  fwrite(obuf, 1, por, stdout);
  por = 0;
}

void rd1(char &c) {
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (c <= ' ');
}

void rd1(string &x) {
  x.clear();
  while (true) {
    if (pil == pir) load();
    while (pil < pir && ibuf[pil] <= ' ') ++pil;
    if (pil < pir) break;
  }
  while (true) {
    uint32_t p = pil;
    while (pil < pir && ibuf[pil] > ' ') ++pil;
    x.append(ibuf + p, pil - p);
    if (pil < pir) {
      ++pil;
      return;
    }
    load();
  }
}

template <typename T>
void rd1_real(T &x) {
  string s;
  rd1(s);

  if constexpr (!is_same_v<T, long double>)
  {
    auto [p, ec] = from_chars(s.data(), s.data() + s.size(), x);
    if (ec == errc{} && p == s.data() + s.size()) return;
  }

  if constexpr (is_same_v<T, long double>)
    x = stold(s);
  else
    x = stod(s);
}

template <bool check_buffer = true, typename T>
void rd1_integer(T &x) {
  using U = unsigned_integer_t<T>;
  bool minus = false;
  U val = 0;
  if constexpr (check_buffer)
    if (pil + 100 > pir) load();
  uint32_t p = pil;
  while (ibuf[p] < '-') ++p;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (ibuf[p] == '-') minus = true, ++p;
  }
  while ('0' <= ibuf[p]) val = val * 10 + (ibuf[p++] & 15);
  pil = p;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (minus)
    {
      const U min_abs = U(numeric_limits<T>::max()) + 1;
      x = val == min_abs ? numeric_limits<T>::lowest() : -T(val);
    }
    else x = T(val);
  }
  else
    x = T(val);
}

void rd1(int &x) { rd1_integer(x); }
void rd1(ll &x) { rd1_integer(x); }
void rd1(i128 &x) { rd1_integer(x); }
void rd1(uint &x) { rd1_integer(x); }
void rd1(ull &x) { rd1_integer(x); }
void rd1(u128 &x) { rd1_integer(x); }
void rd1(double &x) { rd1_real(x); }
void rd1(long double &x) { rd1_real(x); }

template <class... T>
void rd1(tuple<T...> &tpl) {
  apply([](auto &...x) { (rd1(x), ...); }, tpl);
}

template <class T>
void rd1(vc<T> &x) {
  for (auto &d: x) rd1(d);
}

template <class... T>
void read(T &...x) {
  if constexpr (sizeof...(T) <= SIZ / 100 &&
                ((!is_same_v<T, char> &&
                  (is_integral_v<T> || is_same_v<T, i128> || is_same_v<T, u128>)) && ...)) {
    if (pil + 100 * sizeof...(T) > pir) load();
    (rd1_integer<false>(x), ...);
  }
  else
    (rd1(x), ...);
}

void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
void wt1(string_view s) {
  while (!s.empty()) {
    if (por == SIZ) flush();
    size_t n = min<size_t>(s.size(), SIZ - por);
    memcpy(obuf + por, s.data(), n);
    por += n;
    s.remove_prefix(n);
  }
}

template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  using U = unsigned_integer_t<T>;
  U ux;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>)
  {
    if (x < 0)
      obuf[por++] = '-', ux = U(0) - U(x);
    else
      ux = U(x);
  }
  else
    ux = x;
  int outi;
  for (outi = 96; ux >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[ux % 10000], 4);
    ux /= 10000;
  }
  if (ux >= 1000) {
    memcpy(obuf + por, pre.num[ux], 4);
    por += 4;
  } else if (ux >= 100) {
    memcpy(obuf + por, pre.num[ux] + 1, 3);
    por += 3;
  } else if (ux >= 10) {
    int q = (ux * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (ux - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = ux | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

template <typename T>
void wt1_real(T x) {

  if constexpr (!is_same_v<T, long double>)
  {
    auto [p, ec] = to_chars(out, out + sizeof(out), x, chars_format::fixed, 15);
    if (ec == errc{}) {
      wt1(string_view(out, p));
      return;
    }
  }

  ostringstream oss;
  oss << fixed << setprecision(15) << x;
  wt1(oss.str());
}

void wt1(int x) { wt1_integer(x); }
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x);
void wt1(i128 x) { wt1_integer(x); }
void wt1(u128 x) { wt1_integer(x); }
void wt1(double x) { wt1_real(x); }
void wt1(long double x) { wt1_real(x); }

template <class T, class U>
void wt1(const pair<T, U> &val);
template <class... T>
void wt1(const tuple<T...> &tpl);
template <class T, size_t S>
void wt1(const array<T, S> &val);
template <class T>
void wt1(const vector<T> &val);

template <class... T>
void write(T &&...x) {
  (wt1(std::forward<T>(x)), ...);
}

template <class... T>
void print(T &&...x) {
  if constexpr (sizeof...(T))
  {
    int i = 0;
    ((i++ ? wt1(' ') : void(), wt1(std::forward<T>(x))), ...);
  }
  wt1('\n');
}

} 

struct Dummy {
  Dummy() { atexit(fastio::flush); }
} dummy;

namespace internal
{

template <class... Ts>
void READnodump(Ts &...a) { fastio::read(a...); }

template <class... T>
void READVECnodump(int n, vc<T> &...v)
{
  (v.resize(n), ...);
  READnodump(v...);
}

}; 

#define READ(...) internal::READnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define IN(T,...) T __VA_ARGS__; READ(__VA_ARGS__)

#define LL(...) IN(ll, __VA_ARGS__)

#define READVEC(...) internal::READVECnodump(__VA_ARGS__); dump(__VA_ARGS__)

#define VEC(T,n,...) vc<T> __VA_ARGS__; READVEC(n, __VA_ARGS__)

#define PRINT fastio::print

namespace internal
{

template <size_t... I, class A, class B>
auto &tuple_add_impl(A &a, const B &b, const index_sequence<I...>)
{
  ((get<I>(a) += get<I>(b)), ...);
  return a;
}

}; 

template <class... Ts, class Tp>
tuple<Ts...> &operator+=(tuple<Ts...> &a, const Tp &b)
{ return internal::tuple_add_impl(a, b, make_index_sequence<tuple_size_v<tuple<Ts...>>>{}); }

template <class T, class Add>
void offset(vc<T> &v, const Add &add) { for (auto &vi : v) vi += add; }

namespace internal
{

};

// https://github.com/miscalculation53/library/tree/wip/template/template_random.hpp

mt19937_64 mt;

namespace internal
{
}; 

// https://github.com/miscalculation53/library/tree/wip/math/modint/template_modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_static.hpp

// https://github.com/miscalculation53/library/tree/wip/utils/larger_int.hpp

template <class T>
struct larger_int
{
  using type = T;
};

#define LARGER_INT(T,U) template <> struct larger_int<T> { using type = U; };

LARGER_INT(signed char, short)
LARGER_INT(short, int)
LARGER_INT(int, long long)
LARGER_INT(long, __int128_t)
LARGER_INT(long long, __int128_t)

LARGER_INT(unsigned char, unsigned short)
LARGER_INT(unsigned short, unsigned int)
LARGER_INT(unsigned int, unsigned long long)
LARGER_INT(unsigned long, __uint128_t)
LARGER_INT(unsigned long long, __uint128_t)

#undef LARGER_INT

template <class T>
using larger_int_t = typename larger_int<T>::type;
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_isprime.hpp

namespace internal
{

template <class T>
constexpr ll powmod_constexpr(ll x, ll n, T m)
{
  if (m == 1)
    return 0;
  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  U r = 1, y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = L(r) * y % m;
    y = L(y) * y % m;
    n >>= 1;
  }
  return r;
}

template <class T>
constexpr bool isprime_constexpr(T n)
{
  if constexpr (sizeof(T) > 4)
  {
    if (n <= INT_MAX)
      return isprime_constexpr<int>(n);
  }

  if (n <= 1)
    return false;
  if (n == 2 || n == 7 || n == 61)
    return true;
  if (n % 2 == 0)
    return false;

  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;

  using U = make_unsigned_t<T>;
  using L = larger_int_t<U>;

  auto miller_rabin = [&](const auto &bases) constexpr
  {
    for (ll a : bases)
    {
      ll t = d, y = powmod_constexpr(a, t, n);
      while (t != n - 1 && y != 1 && y != n - 1)
      {
        y = L(y) * y % n;
        t <<= 1;
      }
      if (y != n - 1 && t % 2 == 0)
        return false;
    }
    return true;
  };

  if constexpr(sizeof(T) <= 4)
  {
    constexpr ll bases[3] = {2, 7, 61};
    return miller_rabin(bases);
  }
  else
  {
    constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
    return miller_rabin(bases);
  }
}

template <auto n>
constexpr bool isprime = isprime_constexpr(n);

};

namespace internal
{

template <auto M>
struct policy_static
{
  using mod_type = decltype(M);
  using value_type = make_unsigned_t<mod_type>;
  using calc_type = larger_int_t<value_type>;

  static constexpr bool is_prime = isprime_constexpr(M);

  static constexpr mod_type mod() { return M; }
  static constexpr value_type umod() { return M; }

  static constexpr value_type init(value_type v) { return v; }
  static constexpr mod_type val(value_type v) { return v; }
  static constexpr value_type mul(value_type a, value_type b)
  {
    return (value_type)((calc_type(a) * b) % M);
  }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_barrett32.hpp

namespace internal
{

struct barrett32
{
  uint m;
  ull im;
  explicit barrett32(uint m) : m(m), im((ull)(-1) / m + 1) {}
  uint umod() const { return m; }
  uint mul(uint a, uint b) const
  {
    ull z = a;
    z *= b;
    ull x = ull((u128(z) * im) >> 64);
    ull y = x * m;
    return uint(z - y + (z < y ? m : 0));
  }
};

template <int id>
struct policy_barrett32
{
  using value_type = uint;
  using calc_type = ull;
  using mod_type = int;
  
  static constexpr bool is_prime = false;
  static inline barrett32 reducer{998244353};
  static mod_type mod() { return reducer.umod(); }
  static value_type init(value_type v) { return v; }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/modint/modint_internal_montgomery64.hpp

namespace internal
{

inline constexpr ull inv64(ull a)
{
  ull x = a;
  while (a * x != 1) x *= 2 - a * x;
  return x;
}

struct montgomery64odd
{
  ull m, im, sq;
  explicit montgomery64odd(ull m) : m(m), im(inv64(m)), sq(-u128(m) % m) {}
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    auto t = (x + u128(m) * (-im * ull(x))) >> 64;
    if (t >= m) t -= m;
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * sq); }
};

struct montgomery64
{
  ull m, mx, imx, d, q;
  uint b;
  explicit montgomery64(ull m) : m(m)
  {
    b = countr_zero(m), mx = m >> b; 
    imx = inv64(mx);
    d = powmod_constexpr((mx + 1) / 2, b, mx); 
    u128 sq = -u128(mx) % mx; 
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    if (b == 0)
    {
      auto t = (x + u128(mx) * (-imx * ull(x))) >> 64;
      if (t >= m) t -= m;
      return (ull)t;
    }
    ull p = x & MASK(b); 
    x = (x >> b) + p * d;
    ull y = p << (64 - b);
    auto t = (x + u128(mx) * (imx * (y - ull(x)))) >> (64 - b);
    if (t >= m) { t -= m; if (t >= m) t -= m; }
    return (ull)t;
  }
  ull inv_reduce(i128 v) const { return reduce(u128(v % m + m) * q); }
};

template <int id>
struct policy_montgomery64_odd
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64odd reducer{(1LL << 61) - 1};
  static mod_type mod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
};

template <int id>
struct policy_montgomery64
{
  using value_type = ull;
  using calc_type = u128;
  using mod_type = ll;

  static constexpr bool is_prime = false;
  static inline montgomery64 reducer{(1LL << 61) - 1};
  static mod_type mod() { return reducer.umod(); }
  static value_type init(value_type v) { return reducer.inv_reduce(v); }
};

};
// https://github.com/miscalculation53/library/tree/wip/math/extgcd.hpp

template <class T = ll>
constexpr tuple<T, T, T> extgcd(T a, T b)
{
  if (a == 0 && b == 0)
    return {0, 0, 0};
  
  T x1 = 1, y1 = 0, z1 = a;
  T x2 = 0, y2 = 1, z2 = b;
  while (z2 != 0)
  {
    
    T q = z1 / z2;
    tie(x1, x2) = make_pair(x2, x1 - q * x2);
    tie(y1, y2) = make_pair(y2, y1 - q * y2);
    tie(z1, z2) = make_pair(z2, z1 - q * z2);
  }
  if (z1 < 0)
    x1 = -x1, y1 = -y1, z1 = -z1;
  return {z1, x1, y1};
}

namespace internal
{

  template <class Policy>
  struct modint_impl
  {
    using V = typename Policy::value_type;
    using M = typename Policy::mod_type;
    using mint = modint_impl;

  private:
    V _v;

  public:
    static constexpr M mod() { return Policy::mod(); }

    modint_impl() : _v(0) {}

    template <class T, typename = enable_if_t<is_integral_ext<T>>>
    modint_impl(T v)
    {
      V rem;
      if constexpr (is_signed_ext<T>)
      {
        using S = make_signed_t<V>;
        S x = v % S(Policy::umod());
        if (x < 0)
          x += Policy::umod();
        rem = x;
      }
      else
        rem = V(v % Policy::umod());
      _v = Policy::init(rem);
    };

    M val() const { return Policy::val(_v); }

    mint &operator+=(const mint &rhs)
    {
      _v += rhs._v;
      if (_v >= Policy::umod())
        _v -= Policy::umod();
      return *this;
    }
    mint &operator-=(const mint &rhs)
    {
      _v -= rhs._v;
      if (_v >= Policy::umod())
        _v += Policy::umod();
      return *this;
    }
    mint &operator*=(const mint &rhs)
    {
      _v = Policy::mul(_v, rhs._v);
      return *this;
    }

    template <class T>
    mint pow(T n) const
    {
      assert(n >= 0);
      mint x = *this, r = 1;
      while (n)
      {
        if (n & 1)
          r *= x;
        x *= x;
        n >>= 1;
      }
      return r;
    }
    mint inv() const
    {
      if constexpr (Policy::is_prime)
      {
        return pow(mod() - 2);
      }
      else
      {
        auto [g, x, y] = extgcd<M>(val(), mod());
        assert(g == 1);
        return mint(x);
      }
    }

    friend mint operator+(const mint &lhs, const mint &rhs) { return mint(lhs) += rhs; }
    friend mint operator-(const mint &lhs, const mint &rhs) { return mint(lhs) -= rhs; }
    friend mint operator*(const mint &lhs, const mint &rhs) { return mint(lhs) *= rhs; }
    friend mint operator/(const mint &lhs, const mint &rhs) { return mint(lhs) /= rhs; }
    friend bool operator==(const mint &lhs, const mint &rhs) { return lhs._v == rhs._v; }
    friend bool operator!=(const mint &lhs, const mint &rhs) { return lhs._v != rhs._v; }
  
    friend void rd1(mint &x)
    {
      long long a;
      fastio::rd1(a);
      x = a;
    }
    friend void wt1(const mint &x)
    {
      fastio::wt1(x.val());
    }

  };

};

template <int mod>
using static_modint32 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint32 = internal::modint_impl<internal::policy_barrett32<id>>;
template <ll mod>
using static_modint64 = internal::modint_impl<internal::policy_static<mod>>;
template <int id>
using dynamic_modint64_odd = internal::modint_impl<internal::policy_montgomery64_odd<id>>;
template <int id>
using dynamic_modint64 = internal::modint_impl<internal::policy_montgomery64<id>>;

using modint998244353 = static_modint32<998244353>;

template <class T>
struct is_modint : std::false_type
{
};
template <class Policy>
struct is_modint<internal::modint_impl<Policy>> : std::true_type
{
};
template <class T>
inline constexpr bool is_modint_v = is_modint<T>::value;

template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint32<m>> : true_type {};
template <ll m>
struct is_static_modint<static_modint64<m>> : true_type {};
template <class T>
inline constexpr bool is_static_modint_v = is_static_modint<T>::value;

template <class T>
struct is_dynamic_modint : false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint32<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64_odd<id>> : true_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint64<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;

template <typename, typename = void>
struct has_mod : std::false_type
{
};
template <typename T>
struct has_mod<T, std::void_t<decltype(T::mod())>> : std::true_type
{
};

// https://github.com/miscalculation53/library/tree/wip/math/modint/power_table.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/binomial.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/stom.hpp

// https://github.com/miscalculation53/library/tree/wip/math/modint/to_rational.hpp

// https://github.com/miscalculation53/library/tree/wip/math/svp2d.hpp

namespace cpp_dump
{
  struct mint_to_rat_fn
  {
    template <class T>
    constexpr auto operator()(const T &x) const -> decltype(mint_to_rat(x))
    {
      return mint_to_rat(x);
    }
  };

  struct rat_closure : std::ranges::range_adaptor_closure<rat_closure>
  {
    template <typename T>
    constexpr auto operator()(T &&t) const
    {
      if constexpr (!std::ranges::range<T> && std::invocable<mint_to_rat_fn, decltype(std::forward<T>(t))>)
      {
        return mint_to_rat_fn{}(std::forward<T>(t));
      }
      else if constexpr (std::ranges::range<T>)
      {
        using Ref = std::ranges::range_reference_t<T>;

        if constexpr (std::invocable<mint_to_rat_fn, decltype(std::forward<Ref>(std::declval<Ref>()))>)
        {
          return std::forward<T>(t) | std::views::transform(mint_to_rat_fn{});
        }
        else if constexpr (std::ranges::range<Ref>)
        {
          return std::forward<T>(t) | std::views::transform([this](auto &&inner)
                                                            { return (*this)(std::forward<decltype(inner)>(inner)); });
        }
        else
        {
          static_assert(false);
        }
      }
      else
      {
        static_assert(false);
      }
    }
  };

  template <typename T>
    requires(!std::ranges::range<T> && std::invocable<mint_to_rat_fn, T>)

  constexpr rat_closure rat()
  {
    return rat_closure{};
  }
}

using mint = modint998244353;

void init()
{
  oj(mt.seed(random_device()()));
}

// https://github.com/miscalculation53/library/tree/wip/graph/sssp.hpp

// https://github.com/miscalculation53/library/tree/wip/graph/graph.hpp

// https://github.com/miscalculation53/library/tree/wip/ds/csr.hpp

template <class T, bool is_erasable = false>
struct CSR
{
protected:
  int n, m;
  
  vc<int> start;
  vc<T> elist;
  vc<int> len;
  inline int get_last(int i) const
  {
    if constexpr (is_erasable)
      return start[i] + len[i];
    else
      return start[i + 1];
  }

  template <class Iter>
  struct RowBase
  {
    using iterator = Iter;
    using reference = typename iterator_traits<iterator>::reference;

  private:
    iterator begi, endi;

  public:
    RowBase(const iterator &begi, const iterator &endi) : begi(begi), endi(endi) {}
    inline iterator begin() const { return begi; }
    inline iterator end() const { return endi; }

    inline reference operator[](int i) const { return *(begi + i); }

  };
  using Row = RowBase<typename vc<T>::iterator>;
  using ConstRow = RowBase<typename vc<T>::const_iterator>;

public:
  CSR() {}
  
  CSR(const vc<int> &row_sizes)
      : n(row_sizes.size())
  {
    fec(s : row_sizes) assert(s >= 0);
    start = cumlsum(row_sizes);
    m = start.back();
    elist.resize(m);
    if constexpr (is_erasable)
      len = row_sizes;
  }
  
  Row operator[](int i) { return Row(elist.begin() + start[i], elist.begin() + get_last(i)); }
  ConstRow operator[](int i) const
  { return ConstRow(elist.begin() + start[i], elist.begin() + get_last(i)); }

  int offset(int i) const
  {
    assert(0 <= i && i <= n);
    return start[i];
  }

};

template <class Cost>
struct Edge
{
  int from, to;
  Cost cost;
  int index;
  Edge() : from(-1), to(-1), index(-1) {}
  Edge(int s, int t, Cost c, int i = -1) : from(s), to(t), cost(c), index(i) {}
  bool operator<(const Edge &rhs) const;
  
};

template <class Cost>
struct GraphArc
{
  int to, index;
  Cost cost;
  bool operator<(const GraphArc &rhs) const;
};

template <bool is_directed, class Cost, bool is_erasable = false>
struct Graph
{
  using E = Edge<Cost>;
  using A = GraphArc<Cost>;

protected:
  using GE = conditional_t<is_erasable, E, A>;

  int n = 0, m = 0, era = 0;
  CSR<GE, is_erasable> g;
  vc<int> eid_to_elist_id;

  struct OutEdgeIter
  {
    using iterator_category = input_iterator_tag;
    using value_type = E;
    using reference = const E &;

    int from;
    typename vc<GE>::const_iterator it;
    mutable E e;

    bool operator==(const OutEdgeIter &rhs) const;
  };

  template <class F>
  void build(F input_edge)
  {
    vc<int> row_sizes(n);
    repi(i, m)
    {
      auto [u, v, w] = input_edge(i);
      assert(0 <= u && u < n && 0 <= v && v < n);
      row_sizes[u]++;
      if constexpr (!is_directed)
        if (u != v) row_sizes[v]++;
    }
    g = CSR<GE, is_erasable>(row_sizes);
    if constexpr (is_erasable)
      eid_to_elist_id.assign((is_directed ? 1 : 2) * m, -1);

    vc<int> cnt(n);
    repi(i, m)
    {
      auto [u, v, w] = input_edge(i);
      int j = cnt[u]++;
      int k = g.offset(u) + j;
      if constexpr (is_erasable)
        g[u][j] = E(u, v, w, i);
      else
        g[u][j] = A{int(v), i, w};
      if constexpr (is_erasable)
      {
        int id = is_directed ? i : 2 * i + (u <= v);
        eid_to_elist_id[id] = k;
      }
      if constexpr (!is_directed)
      {
        if (u != v)
        {
          j = cnt[v]++;
          k = g.offset(v) + j;
          if constexpr (is_erasable)
            g[v][j] = E(v, u, w, i);
          else
            g[v][j] = A{int(u), i, w};
          if constexpr (is_erasable)
            eid_to_elist_id[2 * i + (v <= u)] = k;
        }
      }
    }
  }

public:
  template <class I>
  Graph(int n, const vc<tuple<I, I, Cost>> &es) : n(n), m(es.size()), era(0)
  {
    build(LMD(i, es[i]));
  }

  template <class I = ll>
  I size() const { return n; }
  
  template <class I = ll>
  I num_of_edges() const { return m - era; }

  auto out_edges(int v) const;
  
  auto out_arcs(int v) const { return g[v]; }
  
  vc<E> edges() const;
  
private:
};

// https://github.com/miscalculation53/library/tree/wip/ds/my_queue.hpp

template <class T>
struct MyQueue
{
private:
  vc<T> d;
  int pos = 0;

public:
  void reserve(int n);
  bool empty() const { return pos == SZ<int>(d); }
  void push(const T &t) { d.eb(t); }
  T &front() { return d[pos]; }
  void pop() { pos++; }

};

template <bool is_directed, class Cost, auto infty = INF>
struct ShortestPath : Graph<is_directed, Cost>
{
  using Graph<is_directed, Cost>::Graph;
  using Graph<is_directed, Cost>::size;
  using Graph<is_directed, Cost>::num_of_edges;
  using Graph<is_directed, Cost>::out_edges;
  using Graph<is_directed, Cost>::out_arcs;
  using Graph<is_directed, Cost>::edges;

private:
  static constexpr decltype(auto) inf() { return resolved_value<Cost, infty>(); }

  vc<Cost> dists;
  vc<Edge<Cost>> prv;
  int source = -1;
  bool solved_all = false;

  void init_solve(int s, int t)
  {
    source = s;
    solved_all = (t == -1);
  }

public:
  vc<Cost> bfs(int s, int t = -1)
  {
    const int n = size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    MyQueue<int> que;
    dists[s] = 0;
    que.push(s);
    while (!que.empty())
    {
      auto v = que.front();
      que.pop();
      if (v == t)
        break;
      fe(e : out_arcs(v))
      {
        if (chmin(dists[e.to], dists[v] + e.cost))
        {
          prv[e.to] = Edge<Cost>(v, e.to, e.cost, e.index);
          que.push(e.to);
        }
      }
    }
    return dists;
  }

  vc<Cost> bfs01(int s, int t = -1)
  {
    const int n = size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    vc<int> cur{int(s)}, nxt;
    vc<unsigned char> used(n, false);
    dists[s] = 0;
    while (!cur.empty() || !nxt.empty())
    {
      if (cur.empty()) cur.swap(nxt);
      int v = cur.back();
      cur.pop_back();
      if (used[v])
        continue;
      used[v] = true;
      if (v == t)
        break;
      fe(e : out_arcs(v))
      {
        if (chmin(dists[e.to], dists[v] + e.cost))
        {
          prv[e.to] = Edge<Cost>(v, e.to, e.cost, e.index);
          if (e.cost == 0)
            cur.eb(e.to);
          else
            nxt.eb(e.to);
        }
      }
    }
    return dists;
  }

  vc<Cost> dial(int s, int max_cost, int t = -1);

  vc<Cost> dijkstra(int s, int t = -1)
  {
    const int n = size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    pql<pair<Cost, int>> pque;
    dists[s] = 0;
    pque.push({0, s});
    while (!pque.empty())
    {
      auto [d, v] = pque.top();
      pque.pop();
      if (v == t)
        break;
      if (dists[v] != d)
        continue;
      fec(e : out_arcs(v))
      {
        Cost nd = dists[v] + e.cost;
        if (chmin(dists[e.to], nd))
        {
          prv[e.to] = Edge<Cost>(v, e.to, e.cost, e.index);
          pque.push({nd, e.to});
        }
      }
    }
    return dists;
  }

  vc<Cost> dijkstra_dense(int s, int t = -1)
  {
    const int n = size();
    assert(0 <= s && s < n);
    init_solve(s, t);
    dists.assign(n, inf());
    prv.assign(n, {});
    vc<bool> ok(n, false);
    dists[s] = 0;
    repi(_, n)
    {
      Cost mn = inf();
      int v = -1;
      repi(u, n) if (!ok[u] && chmin(mn, dists[u])) v = u;
      if (v == -1)
        break;
      if (v == t)
        break;
      ok[v] = true;
      fec(e : out_arcs(v))
      {
        if (chmin(dists[e.to], dists[v] + e.cost))
          prv[e.to] = Edge<Cost>(v, e.to, e.cost, e.index);
      }
    }
    return dists;
  }

  vc<Cost> bellman_ford(int s)
  {
    const int n = size();
    assert(0 <= s && s < n);
    init_solve(s, -1);
    dists.assign(n, inf());
    prv.assign(n, {});
    dists[s] = 0;
    repi(t, 2 * n)
    {
      repi(v, n)
      {
        if (dists[v] == inf())
          continue;
        fec(e : out_arcs(v))
        {
          Cost nd = dists[v] == -inf() ? -inf() : dists[v] + e.cost;
          if (dists[e.to] > nd)
          {
            prv[e.to] = Edge<Cost>(v, e.to, e.cost, e.index);
            if (t == n - 1)
              dists[e.to] = -inf();
            else
              dists[e.to] = nd;
          }
        }
      }
    }
    return dists;
  }

  vc<Cost> solve(int s, int t = -1)
  {
    bool neg = false;
    int zcnt = 0;
    Cost wplus1 = -inf();
    Cost max_cost = 0;
    bool wpluscnt_geq2 = false;
    repi(v, size()) fec(e : out_arcs(v))
    {
      if (e.cost < 0)
      {
        neg = true;
        break;
      }
      chmax(max_cost, e.cost);
    }
    repi(v, size()) fec(e : out_arcs(v))
    {
      if (e.cost == 0)
        zcnt++;
      if (e.cost > 0)
      {
        if (wplus1 < 0)
          wplus1 = e.cost;
        else if (wplus1 != e.cost)
        {
          wpluscnt_geq2 = true;
          break;
        }
      }
    }
    if (neg)
      return bellman_ford(s);
    else if (wpluscnt_geq2)
    {
      const ll n = size(), m = num_of_edges();
      if (n * n < (m << 4))
        return dijkstra_dense(s, t);
      else
      {
        if constexpr (is_integral_ext<Cost>)
        {
          const u128 c = u128(max_cost);
          const ull lg = max<ull>(1, bit_width(ull(n)) - 1);
          if (c < u128(INT_MAX) && u128(n) * c <= u128(m) * lg)
            return dial(s, int(c), t);
        }
        return dijkstra(s, t);
      }
    }
    else
    {
      if (zcnt == 0)
        return bfs(s, t);
      else
        return bfs01(s, t);
    }
  }

};
// https://github.com/miscalculation53/library/tree/wip/math/bigint.hpp

// https://github.com/miscalculation53/library/tree/wip/math/convolution/convolution_ll.hpp

// https://github.com/miscalculation53/library/tree/wip/math/convolution/convolution.hpp

// https://github.com/miscalculation53/library/tree/wip/math/crt.hpp

template <class T = ll, class R0, class R1, class M0, class M1>
constexpr tuple<bool, T, T> crt2(R0 r0_, R1 r1_, M0 m0_, M1 m1_)
{
  T m0 = m0_, m1 = m1_;
  assert(m0 >= 1 && m1 >= 1);
  T r0 = safemod(r0_, m0), r1 = safemod(r1_, m1);
  if (m0 < m1)
    swap(r0, r1), swap(m0, m1);
  if (m0 % m1 == 0)
  {
    if (r0 % m1 != r1)
      return {false, 0, 0};
    return {true, r0, m0};
  }
  auto [g, im, _] = extgcd<T>(m0, m1);
  T u1 = m1 / g;
  if ((r1 - r0) % g)
    return {false, 0, 0};
  T x = (r1 - r0) / g % u1 * im % u1;
  r0 += x * m0;
  m0 *= u1;
  if (r0 < 0)
    r0 += m0;
  return {true, r0, m0};
}

template <class T = ll, class V1, class V2>
constexpr tuple<bool, T, T> crt(const V1 &rs, const V2 &ms)
{
  assert(rs.size() == ms.size());
  const int n = rs.size();
  T r = 0, m = 1;
  repi(i, n)
  {
    auto [ok, nr, nm] = crt2<T>(r, rs[i], m, ms[i]);
    if (!ok)
      return {false, 0, 0};
    r = nr, m = nm;
  }
  return {true, r, m};
}

namespace internal
{

constexpr int primitive_root_constexpr(int m)
{
  if (m == 2)
    return 1;
  if (m == 167772161)
    return 3;
  if (m == 469762049)
    return 3;
  if (m == 754974721)
    return 11;
  if (m == 998244353)
    return 3;
  if (m == 1107296257)
    return 10;
  if (m == 1711276033)
    return 29;
  if (m == 1811939329)
    return 13;
  if (m == 2013265921)
    return 31;
  if (m == 2113929217)
    return 5;
  int divs[20] = {};
  divs[0] = 2;
  int cnt = 1;
  int x = (m - 1) / 2;
  while (x % 2 == 0)
    x /= 2;
  for (int i = 3; (long long)(i)*i <= x; i += 2)
  {
    if (x % i == 0)
    {
      divs[cnt++] = i;
      while (x % i == 0)
      {
        x /= i;
      }
    }
  }
  if (x > 1)
  {
    divs[cnt++] = x;
  }
  for (int g = 2;; g++)
  {
    bool ok = true;
    for (int i = 0; i < cnt; i++)
    {
      if (powmod_constexpr(g, (m - 1) / divs[i], m) == 1)
      {
        ok = false;
        break;
      }
    }
    if (ok)
      return g;
  }
}
template <int m>
constexpr int primitive_root_for_convolution = primitive_root_constexpr(m);

template <class mint, int g = internal::primitive_root_for_convolution<mint::mod()>>
struct fft_info
{
  static constexpr int rank2 = countr_zero(mint::mod() - 1);
  std::array<mint, std::max(3, rank2 + 1)> root;  
  std::array<mint, std::max(3, rank2 + 1)> iroot; 

  std::array<mint, std::max(1, rank2 - 2 + 1)> rate2;
  std::array<mint, std::max(1, rank2 - 2 + 1)> irate2;

  std::array<mint, std::max(1, rank2 - 3 + 1)> rate3;
  std::array<mint, std::max(1, rank2 - 3 + 1)> irate3;

  fft_info()
  {
    root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
    iroot[rank2] = root[rank2].inv();
    for (int i = rank2 - 1; i >= 0; i--)
    {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }

    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 2; i++)
      {
        rate2[i] = root[i + 2] * prod;
        irate2[i] = iroot[i + 2] * iprod;
        prod *= iroot[i + 2];
        iprod *= root[i + 2];
      }
    }
    {
      mint prod = 1, iprod = 1;
      for (int i = 0; i <= rank2 - 3; i++)
      {
        rate3[i] = root[i + 3] * prod;
        irate3[i] = iroot[i + 3] * iprod;
        prod *= iroot[i + 3];
        iprod *= root[i + 3];
      }
    }
  }
};

}  

template <class mint>
bool ntt_ok(int n)
{
  if (n <= 0)
    return false;
  if constexpr (is_static_modint_v<mint>)
  {
    if constexpr (!internal::isprime<mint::mod()>)
      return false;
    static constexpr int rank2 = countr_zero(mint::mod() - 1);
    return n <= (1 << rank2);
  }
  else
    return false;
}

template <int id>
void ntt(vc<dynamic_modint32<id>> &) { assert(false); }

template <auto mod>
void ntt(vc<internal::modint_impl<internal::policy_static<mod>>> &a)
{
  using mint = internal::modint_impl<internal::policy_static<mod>>;
  int n = int(a.size());
  assert(n > 0);
  int h = countr_zero((unsigned int)n);
  assert(n == (1 << h));
  assert(ntt_ok<mint>(n));

  static const internal::fft_info<mint> info;

  int len = 0; 
  while (len < h)
  {
    if (h - len == 1)
    {
      int p = 1 << (h - len - 1);
      mint rot = 1;
      for (int s = 0; s < (1 << len); s++)
      {
        int offset = s << (h - len);
        for (int i = 0; i < p; i++)
        {
          auto l = a[i + offset];
          auto r = a[i + offset + p] * rot;
          a[i + offset] = l + r;
          a[i + offset + p] = l - r;
        }
        if (s + 1 != (1 << len))
          rot *= info.rate2[countr_zero(~(unsigned int)(s))];
      }
      len++;
    }
    else
    {
      
      int p = 1 << (h - len - 2);
      mint rot = 1, imag = info.root[2];
      for (int s = 0; s < (1 << len); s++)
      {
        mint rot2 = rot * rot;
        mint rot3 = rot2 * rot;
        int offset = s << (h - len);
        for (int i = 0; i < p; i++)
        {
          auto mod2 = 1ULL * mint::mod() * mint::mod();
          auto a0 = 1ULL * a[i + offset].val();
          auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
          auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
          auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
          auto a1na3imag =
              1ULL * mint(a1 + mod2 - a3).val() * imag.val();
          auto na2 = mod2 - a2;
          a[i + offset] = a0 + a2 + a1 + a3;
          a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
          a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
          a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
        }
        if (s + 1 != (1 << len))
          rot *= info.rate3[countr_zero(~(unsigned int)(s))];
      }
      len += 2;
    }
  }
}

template <auto mod>
void intt(vc<internal::modint_impl<internal::policy_static<mod>>> &a)
{
  using mint = internal::modint_impl<internal::policy_static<mod>>;
  int n = int(a.size());
  assert(n > 0);
  int h = countr_zero((unsigned int)n);
  assert(n == (1 << h));
  assert(ntt_ok<mint>(n));

  static const internal::fft_info<mint> info;

  int len = h; 
  while (len)
  {
    if (len == 1)
    {
      int p = 1 << (h - len);
      mint irot = 1;
      for (int s = 0; s < (1 << (len - 1)); s++)
      {
        int offset = s << (h - len + 1);
        for (int i = 0; i < p; i++)
        {
          auto l = a[i + offset];
          auto r = a[i + offset + p];
          a[i + offset] = l + r;
          a[i + offset + p] =
              ((unsigned long long)mint::mod() + l.val() - (uint)r.val()) *
              irot.val();
          ;
        }
        if (s + 1 != (1 << (len - 1)))
          irot *= info.irate2[countr_zero(~(unsigned int)(s))];
      }
      len--;
    }
    else
    {
      
      int p = 1 << (h - len);
      mint irot = 1, iimag = info.iroot[2];
      for (int s = 0; s < (1 << (len - 2)); s++)
      {
        mint irot2 = irot * irot;
        mint irot3 = irot2 * irot;
        int offset = s << (h - len + 2);
        for (int i = 0; i < p; i++)
        {
          auto a0 = 1ULL * a[i + offset + 0 * p].val();
          auto a1 = 1ULL * a[i + offset + 1 * p].val();
          auto a2 = 1ULL * a[i + offset + 2 * p].val();
          auto a3 = 1ULL * a[i + offset + 3 * p].val();

          auto a2na3iimag =
              1ULL *
              mint((mint::mod() + a2 - a3) * iimag.val()).val();

          a[i + offset] = a0 + a1 + a2 + a3;
          a[i + offset + 1 * p] =
              (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
          a[i + offset + 2 * p] =
              (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) *
              irot2.val();
          a[i + offset + 3 * p] =
              (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) *
              irot3.val();
        }
        if (s + 1 != (1 << (len - 2)))
          irot *= info.irate3[countr_zero(~(unsigned int)(s))];
      }
      len -= 2;
    }
  }
}

namespace internal
{

template <class mint>
vc<mint> convolution_naive(const vc<mint> &a, const vc<mint> &b)
{
  const int n = a.size(), m = b.size();
  const int cnta = n - count(ALL(a), 0), cntb = m - count(ALL(b), 0);
  vc<mint> c(n + m - 1);
  if ((ll)m * cnta > (ll)n * cntb)
  {
    repi(j, m)
    {
      if (b[j] == 0)
        continue;
      repi(i, n) c[i + j] += a[i] * b[j];
    }
  }
  else
  {
    repi(i, n)
    {
      if (a[i] == 0)
        continue;
      repi(j, m) c[i + j] += a[i] * b[j];
    }
  }
  return c;
}

template <class mint>
vc<mint> convolution_ntt(vc<mint> a, vc<mint> b)
{
  const int n = a.size(), m = b.size();
  const int z = bit_ceil(n + m - 1);
  if (a == b)
  {
    a.resize(z);
    ntt(a);
    repi(i, z) a[i] *= a[i];
  }
  else
  {
    a.resize(z), b.resize(z);
    ntt(a), ntt(b);
    repi(i, z) a[i] *= b[i];
  }
  intt(a);
  mint iz = mint(z).inv();
  fem(ai : a) ai *= iz;
  a.resize(n + m - 1);
  return a;
}

template <size_t j, int mod, class T, size_t k>
void convolution_crt_helper(const vc<T> &a, const vc<T> &b, vc<array<T, k>> &cs)
{
  using mint = static_modint32<mod>;
  const int n = a.size(), m = b.size();
  auto c = convolution_ntt(vc<mint>(ALL(a)), vc<mint>(ALL(b)));
  repi(i, n + m - 1) cs[i][j] = c[i].val();
}

template <int ...ms, class T>
vc<T> convolution_crt(const vc<T> &a, const vc<T> &b)
{
  const int n = a.size(), m = b.size();

  constexpr size_t k = sizeof...(ms);
  vc<array<T, k>> cs(n + m - 1);
  constexpr array<int, k> ms_arr = {ms...};
  [&]<size_t... Is>(index_sequence<Is...>)
  {
    (convolution_crt_helper<Is, ms_arr[Is], T, k>(a, b, cs), ...);
  }(make_index_sequence<k>{});

  vc<T> c(n + m - 1);
  repi(i, n + m - 1) c[i] = get<1>(crt(cs[i], ms_arr));
  return c;
}

}  

vc<ll> convolution_4e18(const vc<ll> &a, const vc<ll> &b)
{
  if (a.empty() || b.empty())
    return {};
  const int n = a.size(), m = b.size();
  const int cnta = n - count(ALL(a), 0), cntb = m - count(ALL(b), 0);
  int z = bit_ceil(n + m - 1);
  ll naive_work = min(ll(cnta) * m, ll(cntb) * n);
  ll ntt_work = ll(z) * max(1, int(countr_zero((unsigned)z)));
  if (naive_work <= 18 * ntt_work)
    return internal::convolution_naive(a, b);
  return internal::convolution_crt<2013265921, 2113929217>(a, b);
}
// https://github.com/miscalculation53/library/tree/wip/utils/make_unsigned_ext.hpp

template <class T>
struct make_unsigned_ext
{
  using type = make_unsigned_t<T>;
};
template <>
struct make_unsigned_ext<i128>
{
  using type = u128;
};
template <class T>
using make_unsigned_ext_t = typename make_unsigned_ext<T>::type;

template <class T>
struct make_signed_ext
{
  using type = make_signed_t<T>;
};
template <>
struct make_signed_ext<u128>
{
  using type = i128;
};

template <int base = 10, int digit = 6>
struct BigInteger
{
private:
  static constexpr int BASE = ipow(base, digit);
  vl vec;
  bool is_nega = false;

  static char digit_to_char(int x) { return x < 10 ? char('0' + x) : char('A' + x - 10); }

  void zero_suppress()
  {
    while (!vec.empty() && vec.back() == 0)
      vec.pop_back();
    if (vec.empty())
      is_nega = false;
  }
  void carry(int d = -1)
  {
    const int n = vec.size();
    if (n == 0)
      return;
    if (d < 0)
      d = n - 1;
    repi(i, d)
    {
      vec[i + 1] += vec[i] / BASE;
      vec[i] = vec[i] % BASE;
    }
    for (int i = d; !(0 <= vec[i] && vec[i] < BASE); i++)
    {
      vec.eb(vec[i] / BASE);
      vec[i] = vec[i] % BASE;
    }
  }
  void borrow(int si, int mx)
  {
    ll bor = 0;
    repi(i, si, vec.size())
    {
      ll val = vec[i] - bor;
      if (val < 0)
      {
        bor = (-val + BASE - 1) / BASE;
        vec[i] = val + bor * BASE;
      }
      else
      {
        vec[i] = val;
        bor = 0;
        if (i >= mx)
          break;
      }
    }
    if (bor > 0)
    {
      is_nega ^= 1;
      ll car = 1;
      repi(i, vec.size())
      {
        ll val = (BASE - 1) - vec[i] + car;
        if (val >= BASE)
          vec[i] = val - BASE, car = 1;
        else
          vec[i] = val, car = 0;
      }
    }
    zero_suppress();
    if (vec.empty())
      is_nega = false;
  }

  BigInteger &operator<<=(size_t k)
  {
    vec.insert(vec.begin(), k, 0);
    return *this;
  }
  BigInteger &operator>>=(size_t k)
  {
    vec.erase(vec.begin(), vec.begin() + min(vec.size(), k));
    return *this;
  }
  BigInteger operator<<(size_t k) const { return BigInteger(*this) <<= k; }
  BigInteger operator>>(size_t k) const { return BigInteger(*this) >>= k; }

  pair<BigInteger, int> divmod_digit(int d) const
  {
    BigInteger q = *this;
    ll rem = 0;
    repi(i, SZ(q.vec) - 1, -1, -1)
    {
      ll cur = q.vec[i] + rem * BASE;
      q.vec[i] = cur / d;
      rem = cur % d;
    }
    q.zero_suppress();
    return {q, int(rem)};
  }

  pair<BigInteger, BigInteger> divmod_knuth(const BigInteger &b) const
  {
    BigInteger q, r;
    vl u = vec, v = b.vec;
    const ll d = BASE / (v.back() + 1);
    auto normalize = [&](vl &a)
    {
      ll car = 0;
      fem(x : a)
      {
        ll cur = x * d + car;
        x = cur % BASE, car = cur / BASE;
      }
      if (car) a.eb(car);
    };
    normalize(u), normalize(v);
    const int n = v.size();
    u.eb(0);
    const int m = SZ(u) - n - 1;
    q.vec.resize(m + 1);
    repi(j, m, -1, -1)
    {
      ll cur = u[j + n] * BASE + u[j + n - 1];
      ll x = cur / v[n - 1], rem = cur % v[n - 1];
      if (x == BASE)
        x--, rem += v[n - 1];
      while (n >= 2 && rem < BASE && x * v[n - 2] > rem * BASE + u[j + n - 2])
        x--, rem += v[n - 1];

      ll bor = 0;
      repi(i, n)
      {
        ll z = x * v[i] + bor;
        bor = z / BASE, z %= BASE;
        if (u[j + i] < z)
          u[j + i] += BASE, bor++;
        u[j + i] -= z;
      }
      u[j + n] -= bor;
      if (u[j + n] < 0)
      {
        x--;
        ll car = 0;
        repi(i, n)
        {
          ll z = u[j + i] + v[i] + car;
          u[j + i] = z % BASE, car = z / BASE;
        }
        u[j + n] += car;
      }
      q.vec[j] = x;
    }

    r.vec.assign(u.begin(), u.begin() + n);
    ll rem = 0;
    repi(i, n - 1, -1, -1)
    {
      ll cur = r.vec[i] + rem * BASE;
      r.vec[i] = cur / d, rem = cur % d;
    }
    q.zero_suppress(), r.zero_suppress();
    return {q, r};
  }

  pair<BigInteger, BigInteger> divmod_abs(const BigInteger &b) const
  {
    BigInteger a = abs(), bb = b.abs();
    if (a < bb)
      return {0, a};
    ll work = ll(bb.vec.size()) * (SZ(a.vec) - SZ(bb.vec) + 1);
    if (work <= 600'000)
      return a.divmod_knuth(bb);
    BigInteger ib = bb.inv(a.vec.size());
    BigInteger q = (a * ib) >> (SZ(a.vec) + SZ(bb.vec) - 1);
    BigInteger r = a - bb * q;
    assert(r >= 0);
    if (r >= bb)
      q += 1, r -= bb;
    return {q, r};
  }

  int cmp(const BigInteger &b) const
  {
    if (is_nega ^ b.is_nega)
      return is_nega ? -1 : 1;
    if (vec.size() != b.vec.size())
      return ((vec.size() < b.vec.size()) ^ is_nega) ? -1 : 1;
    repi(i, SZ(vec) - 1, -1, -1)
    {
      if (vec[i] != b.vec[i])
        return ((vec[i] < b.vec[i]) ^ is_nega) ? -1 : 1;
    }
    return 0;
  }

  template <class T>
  void add_abs(int q, T val)
  {
    if (q >= SZ(vec))
      vec.resize(q + 1);
    int i = q;
    while (val > 0)
    {
      if (i >= SZ(vec))
        vec.eb(0);
      val += vec[i];
      vec[i] = ll(val % BASE);
      val /= BASE;
      i++;
    }
  }
  template <class T>
  void sub_abs(int q, T val)
  {
    if (q >= SZ(vec))
      vec.resize(q + 1);
    int i = q;
    while (val > 0)
    {
      if (i >= SZ(vec))
        vec.eb(0);
      vec[i] -= ll(val % BASE);
      val /= BASE;
      i++;
    }
    borrow(q, i - 1);
  }

public:
  BigInteger() {}
  template <class T, typename = enable_if_t<is_integral_ext<T>>>
  BigInteger(T x)
  {
    using U = make_unsigned_ext_t<T>;
    U ux = x;
    if constexpr (is_signed_ext<T>)
    {
      if (x < 0)
        is_nega = true, ux = -ux;
      else
        is_nega = false;
    }
    else
      is_nega = false;
    
    if (ux == 0)
      return;
    while (ux > 0)
    {
      vec.eb((ll)(ux % BASE));
      ux /= BASE;
    }
  }

  string to_string() const
  {
    const int n = vec.size();
    if (n == 0)
      return "0";
    string s;
    s.reserve(size_t(n) * digit + is_nega);
    if (is_nega)
      s += '-';
    ll x = vec.back();
    char buf[digit];
    int len = 0;
    do
    {
      buf[len++] = digit_to_char(x % base);
      x /= base;
    } while (x);
    repi(i, len - 1, -1, -1) s += buf[i];
    repi(i, SZ(vec) - 2, -1, -1)
    {
      size_t p = s.size();
      s.resize(p + digit);
      x = vec[i];
      repi(j, digit - 1, -1, -1)
      {
        s[p + j] = digit_to_char(x % base);
        x /= base;
      }
    }
    return s;
  }

  bool operator<(const BigInteger &b) const { return cmp(b) < 0; }
  bool operator>(const BigInteger &b) const { return cmp(b) > 0; }
  bool operator>=(const BigInteger &b) const { return cmp(b) >= 0; }
  bool operator==(const BigInteger &b) const { return is_nega == b.is_nega && vec == b.vec; }
  bool operator!=(const BigInteger &b) const { return !(*this == b); }

  BigInteger operator-() const
  {
    BigInteger res(*this);
    if (!res.vec.empty())
      res.is_nega ^= 1;
    return res;
  }
  BigInteger abs() const
  {
    BigInteger res(*this);
    res.is_nega = false;
    return res;
  }
  BigInteger &operator+=(const BigInteger &b)
  {
    if (b.vec.empty())
      return *this;
    if (vec.size() < b.vec.size())
      vec.resize(b.vec.size());
    if (is_nega == b.is_nega)
    {
      repi(i, b.vec.size()) vec[i] += b.vec[i];
      carry();
    }
    else
    {
      repi(i, b.vec.size()) vec[i] -= b.vec[i];
      borrow(0, SZ(b.vec) - 1);
    }
    return *this;
  }
  BigInteger &operator-=(const BigInteger &b)
  {
    if (b.vec.empty())
      return *this;
    if (vec.size() < b.vec.size())
      vec.resize(b.vec.size());
    if (is_nega != b.is_nega)
    {
      repi(i, b.vec.size()) vec[i] += b.vec[i];
      carry();
    }
    else
    {
      repi(i, b.vec.size()) vec[i] -= b.vec[i];
      borrow(0, SZ(b.vec) - 1);
    }
    return *this;
  }
  BigInteger &operator*=(const BigInteger &b)
  {
    if (this->vec.empty() || b.vec.empty())
    {
      this->vec.clear();
      this->is_nega = false;
      return *this;
    }
    bool neg = is_nega ^ b.is_nega;
    if (b.vec.size() == 1)
    {
      *this *= int(b.vec[0]);
      is_nega = neg;
      return *this;
    }
    if (vec.size() == 1)
    {
      int v = vec[0];
      *this = b;
      *this *= v;
      is_nega = neg;
      return *this;
    }
    vec = convolution_4e18(this->vec, b.vec);
    carry();
    zero_suppress();
    is_nega = neg;
    return *this;
  }
  BigInteger operator+(const BigInteger &b) const { return BigInteger(*this) += b; }
  BigInteger operator-(const BigInteger &b) const { return BigInteger(*this) -= b; }
  BigInteger operator*(const BigInteger &b) const { return BigInteger(*this) *= b; }
  BigInteger inv(int d) const
  {
    assert(!vec.empty());
    BigInteger a(abs()), c, c2;
    BigInteger b = binsearch(LMD(m, ((a * m).vec.size() <= a.vec.size())), 0, BASE + 1, false, false).first;
    const BigInteger ONE(1), TWO(2);
    for (int k = 1;; k = min(2 * k, d))
    {
      c = a * b;
      if (SZ(b.vec) >= d + 1 && SZ(c.vec) == SZ(a.vec) + SZ(b.vec) - 1)
      {
        c2 = c + a;
        if (c2.vec.size() == a.vec.size() + b.vec.size())
        {
          if (c2 == ONE << (SZ(c2.vec) - 1))
            b += 1;
          break;
        }
      }
      b *= (TWO << (SZ(a.vec) + SZ(b.vec) - 1)) - c;
      if (SZ(b.vec) >= k + 1)
        b >>= SZ(b.vec) - k - 1;
    }
    b >>= 1;
    if (is_nega)
      b = -b;
    return b;
  }
  BigInteger operator/(const BigInteger &b) const
  {
    assert(!b.vec.empty());
    if (b.vec.size() == 1)
    {
      BigInteger q = divmod_digit(int(b.vec[0])).first;
      q.is_nega = !q.vec.empty() && (is_nega ^ b.is_nega);
      return q;
    }
    BigInteger q = divmod_abs(b).first;
    q.is_nega = !q.vec.empty() && (is_nega ^ b.is_nega);
    return q;
  }
  pair<BigInteger, BigInteger> divmod(const BigInteger &b) const
  {
    assert(!b.vec.empty());
    if (b.vec.size() == 1)
    {
      auto [q, rem] = divmod_digit(int(b.vec[0]));
      q.is_nega = !q.vec.empty() && (is_nega ^ b.is_nega);
      BigInteger r(rem);
      r.is_nega = rem && is_nega;
      return {q, r};
    }
    auto [q, r] = divmod_abs(b);
    q.is_nega = !q.vec.empty() && (is_nega ^ b.is_nega);
    r.is_nega = !r.vec.empty() && is_nega;
    return {q, r};
  }
  BigInteger operator%(const BigInteger &b) const { return divmod(b).second; }
  BigInteger &operator%=(const BigInteger &b) { return *this = *this % b; }

  template <class T, typename = enable_if_t<is_integral_ext<T>>>
  BigInteger &operator+=(T v)
  {
    if (v == 0)
      return *this;
    bool v_nega = false;
    using U = make_unsigned_ext_t<T>;
    U uv = v;
    if constexpr (is_signed_ext<T>)
    {
      if (v < 0)
        v_nega = true, uv = -uv;
    }
    using V = larger_int_t<U>;
    if (is_nega == v_nega)
      add_abs(0, (V)uv);
    else
      sub_abs(0, (V)uv);
    return *this;
  }
  template <class T, typename = enable_if_t<is_integral_ext<T>>>
  BigInteger &operator*=(T v)
  {
    if (vec.empty() || v == 0)
    {
      vec.clear();
      is_nega = false;
      return *this;
    }
    bool v_nega = false;
    using U = make_unsigned_ext_t<T>;
    U uv = v;
    if constexpr (is_signed_ext<T>)
    {
      if (v < 0)
        v_nega = true, uv = -uv;
    }
    is_nega ^= v_nega;
    using V = larger_int_t<U>;
    V car = 0;
    repi(i, SZ(vec))
    {
      car += (V)vec[i] * uv;
      vec[i] = (ll)(car % BASE);
      car /= BASE;
    }
    while (car > 0)
    {
      vec.eb((ll)(car % BASE));
      car /= BASE;
    }
    return *this;
  }
  template <class T, typename = enable_if_t<is_integral_ext<T>>>
  BigInteger &operator/=(T v)
  {
    assert(v != 0);
    if (vec.empty())
      return *this;
    bool v_nega = false;
    using U = make_unsigned_ext_t<T>;
    U uv = v;
    if constexpr (is_signed_ext<T>)
    {
      if (v < 0)
        v_nega = true, uv = -uv;
    }
    is_nega ^= v_nega;
    using V = larger_int_t<U>;
    V rem = 0;
    repi(i, SZ(vec) - 1, -1, -1)
    {
      V cur = vec[i] + rem * BASE;
      vec[i] = (ll)(cur / uv);
      rem = cur % uv;
    }
    zero_suppress();
    return *this;
  }

  // base**i * coef を加算、ならし O(1) 時間
  // base**i * coef を減算、数が常に非負の場合はならし O(1) 時間

  template <class T, typename = enable_if_t<is_integral_ext<T>>>
  BigInteger operator*(T v) const { return BigInteger(*this) *= v; }
  template <class T, typename = enable_if_t<is_integral_ext<T>>>
  BigInteger operator/(T v) const { return BigInteger(*this) /= v; }
};

template <class T, int base, int digit, typename = enable_if_t<is_integral_ext<T>>>
BigInteger<base, digit> operator*(T a, const BigInteger<base, digit> &b)
{
  return b * a;
}

template <int base, int digit>
BigInteger<base, digit> gcd(BigInteger<base, digit> a, BigInteger<base, digit> b)
{
  a = a.abs(), b = b.abs();
  while (b != 0)
  {
    a %= b;
    swap(a, b);
  }
  return a;
}

template <int base, int digit>
BigInteger<base, digit> lcm(BigInteger<base, digit> a, BigInteger<base, digit> b)
{
  if (a == 0 || b == 0)
    return 0;
  a = a.abs(), b = b.abs();
  return a / gcd(a, b) * b;
}



template <int base, int digit, class T, enable_if_t<is_integral_ext<T>, int> = 0>
BigInteger<base, digit> lcm(BigInteger<base, digit> a, T b)
{
  return lcm(move(a), BigInteger<base, digit>(b));
}

template <class T, int base, int digit, enable_if_t<is_integral_ext<T>, int> = 0>
BigInteger<base, digit> lcm(T a, BigInteger<base, digit> b)
{
  return lcm(BigInteger<base, digit>(a), move(b));
}

template <int base, int digit>
void wt1(const BigInteger<base, digit> &a)
{
  fastio::wt1(a.to_string());
}





// https://github.com/miscalculation53/library/tree/wip/math/rational.hpp



/**
 * @brief 約分しない有理数
 * @docs docs/math/rational.md
 */

template <class T, class = void>
struct is_rational_ordered : false_type
{};

template <class T>
struct is_rational_ordered<T, void_t<decltype(declval<const T &>() < declval<const T &>())>> : true_type
{};

template <class T>
inline constexpr bool is_rational_ordered_v = is_rational_ordered<T>::value;

template <class T, class = void>
struct rational_has_mod : false_type
{};

template <class T>
struct rational_has_mod<T, void_t<decltype(declval<T>() % declval<T>())>> : true_type
{};

template <class T>
struct Rational
{
  using value_type = T;

  T num, den;

private:


public:




  bool is_infinite() const;

  pair<T, T> reduced() const;







  friend Rational operator+(Rational lhs, const Rational &rhs) { return lhs += rhs; }

  friend Rational operator-(Rational lhs, const Rational &rhs) { return lhs -= rhs; }

  friend Rational operator*(Rational lhs, const Rational &rhs) { return lhs *= rhs; }

  friend Rational operator/(Rational lhs, const Rational &rhs) { return lhs /= rhs; }

  friend bool operator==(const Rational &lhs, const Rational &rhs)
  {
    if (lhs.is_infinite() || rhs.is_infinite())
      return lhs.is_infinite() && rhs.is_infinite() && lhs.num == rhs.num;
    using C = larger_int_t<T>;
    return C(lhs.num) * C(rhs.den) == C(rhs.num) * C(lhs.den);
  }

  friend bool operator!=(const Rational &lhs, const Rational &rhs) { return !(lhs == rhs); }


  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator<(const Rational &lhs, const Rational &rhs)
  {
    if (lhs.is_infinite() || rhs.is_infinite())
    {
      if (lhs.is_infinite() && rhs.is_infinite())
        return lhs.num < rhs.num;
      if (lhs.is_infinite())
        return lhs.num < T(0);
      return T(0) < rhs.num;
    }
    using C = larger_int_t<T>;
    return C(lhs.num) * C(rhs.den) < C(rhs.num) * C(lhs.den);
  }

  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator>(const Rational &lhs, const Rational &rhs) { return rhs < lhs; }

  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator<=(const Rational &lhs, const Rational &rhs) { return !(rhs < lhs); }

  template <class U = T, enable_if_t<is_rational_ordered_v<U>, int> = 0>
  friend bool operator>=(const Rational &lhs, const Rational &rhs) { return !(lhs < rhs); }

  friend ostream &operator<<(ostream &os, const Rational &x)
  {
    auto [num, den] = x.reduced();
    return os << num << '/' << den;
  }


  friend void wt1(const Rational &x)
  {
    auto [num, den] = x.reduced();
    using fastio::wt1;
    wt1(num), wt1('/'), wt1(den);
  }

};

template <class T>
struct is_rational : false_type
{};

template <class T>
struct is_rational<Rational<T>> : true_type
{};

template <class T>
inline constexpr bool is_rational_v = is_rational<T>::value;

namespace std
{
template <class T>
struct numeric_limits<::Rational<T>> : numeric_limits<T>
{
  static constexpr bool is_specialized = numeric_limits<T>::is_specialized;
  static constexpr bool has_infinity = ::is_rational_ordered_v<T>;
  static constexpr bool is_signed = numeric_limits<T>::is_signed;
  static constexpr bool is_integer = false;
  static constexpr bool is_exact = numeric_limits<T>::is_exact;
};
} // namespace std





void main2()
{
  static const BigInteger L = []()
  {
    BigInteger L = 1;
    rep(i, 1, 300) L = lcm(L, i);
    return L;
  }();
  dump(L);

  LL(N, M);
  VEC(tllll, M, UVAB);
  offset(UVAB, tllll{-1, -1, 0, 0});
  vc<tuple<ll, ll, BigInteger<>>> UVW;
  fec([ u, v, a, b ] : UVAB) UVW.eb(u, v, a * (L / b));
  ShortestPath<false, BigInteger<>, []()
               { return L * 300 * 30000; }>
      G(N, UVW);
  auto res = G.solve(0);
  rep(i, 1, N)
  {
    auto num = res[i], den = L;
    auto g = gcd(num, den);
    PRINT(num / g, den / g);
  }
}

void test()
{

}

// https://github.com/miscalculation53/library/tree/wip/template/template_main.hpp


template <auto init, auto main2, auto test>
struct Main
{
  Main()
  {
    cauto CERR = [](string val, string color)
    {
      string s = "\033[" + color + "m" + val + "\033[m";
      


      /* コードテストで確認する際にコメントアウトを外す
      cerr << val;
      //*/
    };
  
    
    CERR("\n[FAST_IO]\n\n", "32");
    
    




    cout << fixed << setprecision(20);
  
    init();
    


  
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
int main() {}