#define INF 4'000'000'000'000'000'037LL
#include <bits/stdc++.h>
using namespace std;
namespace {
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using pll = pair<ll, ll>;
#define vc vector
template <class T>
using vvc = vc<vc<T>>;
using vpll = vc<pll>;
#ifdef __SIZEOF_INT128__
using i128 = __int128_t;
using u128 = __uint128_t;
#endif
#define cauto const auto
#define overload4(_1, _2, _3, _4, name, ...) name
#define rep1(i, n) for (ll i = 0, nnnnn = ll(n); i < nnnnn; 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 repi(...) overload4(__VA_ARGS__, repi3, repi2, repi1)(__VA_ARGS__)
#define fec(...) for (cauto &__VA_ARGS__)
#define fem(...) for (auto &__VA_ARGS__)
template <class T, class U>
inline bool chmin(T &a, U b) { return a > b ? a = b, true : false; }
template <class T = ll, class U, class 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>
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;
}
template <class T = ll, class A, class B, class M>
T mul_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (b == 0)
    return 0;
  return T(a) > T(m) / T(b) ? T(m) : T(a) * T(b);
}
template <class T = ll, class A, class B>
T mul_limited(A a, B b) { return mul_limited<T>(a, b, INF); }
template <class T = ll, class A, class B, class M>
T pow_limited(A a, B b, M m)
{
  assert(a >= 0 && b >= 0 && m >= 0);
  if (a <= 1 || b == 0)
    return min(ipow<T>(a, b), T(m));
  T res = 1, tmp = a;
  while (true)
  {
    if (b & 1)
    {
      if (res > T(m) / tmp)
        return m;
      res *= tmp;
    }
    b >>= 1;
    if (b == 0)
      break;
    if (tmp > T(m) / tmp)
      return m;
    tmp *= tmp;
  }
  return res;
}
template <class T = ll, class A, class B>
T pow_limited(A a, B b) { return pow_limited<T>(a, b, INF); }
#define ALL(a) (a).begin(), (a).end()
#define eb emplace_back
#define LMD(x, fx) ([&](auto x) { return fx; })
template <class T, size_t d, size_t i = 0, class V>
auto dvec(const V (&sz)[d], const T &init)
{
  if constexpr (i < d)
    return vc(sz[i], dvec<T, d, i + 1>(sz, init));
  else
    return init;
}
template <class T, class U>
vc<T> permuted(const vc<T> &a, const vc<U> &p)
{
  const int n = p.size();
  vc<T> res(n);
  repi(i, n)
  {
    assert(0 <= p[i] && p[i] < U(a.size()));
    res[i] = a[p[i]];
  }
  return res;
}
template <class T, class U, class... Ts>
vc<T> permuted(const vc<T> &p, const vc<U> &q, const vc<Ts> &...rs)
{
  return permuted(permuted(p, q), rs...);
}
#if __cplusplus < 202002L
#else
#endif
template <class V>
void unique(V &v) { v.erase(std::unique(ALL(v)), v.end()); }
template <class V, class U>
void rotate(V &v, U k)
{ 
  const U n = v.size();
  k = (k % n + n) % n;
  std::rotate(v.begin(), v.begin() + k, v.end());
}
const vpll DRULgrid = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
const vpll DRULplane = {{0, -1}, {1, 0}, {0, 1}, {-1, 0}};
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;
#if __cplusplus < 202002L
struct identity
{
  template <class T>
  constexpr T &&operator()(T &&t) const noexcept
  { return forward<T>(t); }
};
namespace internal
{
  template <class T = ll, class V, class Judge>
  inline T bound_helper(const V &v, Judge judge)
  {
    int l = -1, r = v.size();
    while (r - l > 1)
    {
      int m = (l + r) / 2;
      if (judge(m))
        l = m;
      else
        r = m;
    }
    return r;
  }
};
#else
#endif
template <class T>
inline constexpr ull MASK(T k) { return (1ULL << k) - 1ULL; }
#if __cplusplus < 202002L
inline constexpr ull bit_width(ull x) { return x == 0 ? 0 : 64 - __builtin_clzll(x); }
inline constexpr ull bit_ceil(ull x) { return x == 0 ? 1ULL : 1ULL << bit_width(x - 1); }
inline constexpr ull countr_zero(ull x) { assert(x != 0); return __builtin_ctzll(x); }
#else
inline constexpr ll bit_width(ll x) { return std::bit_width((ull)x); }
inline constexpr ll bit_floor(ll x) { return std::bit_floor((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); }
inline constexpr ll popcount(ll x) { return std::popcount((ull)x); }
inline constexpr bool has_single_bit(ll x) { return std::has_single_bit((ull)x); }
#endif
#define dump(...)
namespace fastio {
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;
}
template <typename T>
void rd1_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  x = 0;
  while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (minus) x = -x;
  }
}
void rd1(ll &x) { rd1_integer(x); }
template <class T, class U>
void rd1(pair<T, U> &p) {
  return rd1(p.first), rd1(p.second);
}
template <size_t N = 0, typename T>
void rd1_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd1(x);
    rd1_tuple<N + 1>(t);
  }
}
template <class... T>
void rd1(tuple<T...> &tpl) {
  rd1_tuple(tpl);
}
template <size_t N = 0, typename T>
void rd1(array<T, N> &x) {
  for (auto &d: x) rd1(d);
}
template <class T>
void rd1(vc<T> &x) {
  for (auto &d: x) rd1(d);
}
void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd1(h), read(t...);
}
void wt1(const char c) {
  if (por == SIZ) flush();
  obuf[por++] = c;
}
template <typename T>
void wt1_integer(T x) {
  if (por > SIZ - 100) flush();
  if (x < 0) { obuf[por++] = '-', x = -x; }
  int outi;
  for (outi = 96; x >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[x % 10000], 4);
    x /= 10000;
  }
  if (x >= 1000) {
    memcpy(obuf + por, pre.num[x], 4);
    por += 4;
  } else if (x >= 100) {
    memcpy(obuf + por, pre.num[x] + 1, 3);
    por += 3;
  } else if (x >= 10) {
    int q = (x * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (x - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = x | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}
void wt1(int x) { wt1_integer(x); }
template <class T, enable_if_t<is_integral_v<T>, int> = 0>
void wt1(T x) { wt1_integer(x); }
template <class T, class U>
void wt1(const pair<T, U> &val) {
  wt1(val.first);
  wt1(' ');
  wt1(val.second);
}
template <size_t N = 0, typename T>
void wt1_tuple(const T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt1(' '); }
    const auto x = std::get<N>(t);
    wt1(x);
    wt1_tuple<N + 1>(t);
  }
}
template <class... T>
void wt1(const tuple<T...> &tpl) {
  wt1_tuple(tpl);
}
template <class T, size_t S>
void wt1(const array<T, S> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
template <class T>
void wt1(const vector<T> &val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt1(' ');
    wt1(val[i]);
  }
}
void print() { wt1('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt1(head);
  if (sizeof...(Tail)) wt1(' ');
  print(forward<Tail>(tail)...);
}
} // namespace fastio
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);
}
template <class T, class... Ts>
void READVECnodump(int n, vc<T> &v, vc<Ts> &...vs)
{ READVECnodump(n, v), READVECnodump(n, vs...); }
template <class T>
void READVEC2nodump(int n, int m, vvc<T> &v)
{
  v.assign(n, vc<T>(m));
  READnodump(v);
}
template <class T, class... Ts>
void READVEC2nodump(int n, int m, vvc<T> &v, vvc<Ts> &...vs)
{ READVEC2nodump(n, m, v), READVEC2nodump(n, m, vs...); }
template <class T>
void READJAGnodump(int n, vvc<T> &v)
{
  v.resize(n);
  repi(i, n)
  {
    int k;
    READnodump(k);
    READVECnodump(k, v[i]);
  }
}
template <class T, class... Ts>
void READJAGnodump(int n, vvc<T> &v, vvc<Ts> &...vs)
{ READJAGnodump(n, v), READJAGnodump(n, vs...); }
}; // namespace internal
#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
template <class T, class U, class P>
pair<T, U> operator+=(pair<T, U> &a, const P &b)
{
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <class T, class U, class P>
pair<T, U> operator+(pair<T, U> &a, const P &b) { return a += b; }
template <class T, size_t n, class A>
array<T, n> operator+=(array<T, n> &a, const A &b)
{
  for (size_t i = 0; i < n; i++)
    a[i] += b[i];
  return a;
}
template <class T, size_t n, class A>
array<T, n> operator+(array<T, n> &a, const A &b) { return a += b; }
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;
}
}; // namespace internal
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... Ts, class Tp>
tuple<Ts...> operator+(tuple<Ts...> &a, const Tp &b) { return a += b; }
template <class T, const size_t m>
vc<array<T, m>> top(const array<vc<T>, m> &tv)
{
  if (tv.empty()) return {};
  const size_t n = tv[0].size();
  vc<array<T, m>> vt(n);
  for (size_t j = 0; j < m; j++)
  {
    assert(tv[j].size() == n);
    for (size_t i = 0; i < n; i++)
      vt[i][j] = tv[j][i];
  }
  return vt;
}
template <class T, class U>
pair<vc<T>, vc<U>> top(const vc<pair<T, U>> &vt)
{
  const size_t n = vt.size();
  pair<vc<T>, vc<U>> tv;
  tv.first.resize(n), tv.second.resize(n);
  for (size_t i = 0; i < n; i++)
    tie(tv.first[i], tv.second[i]) = vt[i];
  return tv;
}
template <class T, class U>
vc<pair<T, U>> top(const pair<vc<T>, vc<U>> &tv)
{
  const size_t n = tv.first.size();
  assert(n == tv.second.size());
  vc<pair<T, U>> vt(n);
  for (size_t i = 0; i < n; i++)
    vt[i] = make_pair(tv.first[i], tv.second[i]);
  return vt;
}
namespace internal
{
template <size_t... I, class V, class Tp>
auto vt_to_tv_impl(V &tv, const Tp &t, index_sequence<I...>, size_t index)
{ ((get<I>(tv)[index] = get<I>(t)), ...); }
template <size_t... I, class Tp>
auto tv_to_vt_impl(const Tp &tv, index_sequence<I...>, size_t index)
{ return make_tuple(get<I>(tv)[index]...); }
};
template <class... Ts>
auto top(const vc<tuple<Ts...>> &vt)
{
  const size_t n = vt.size();
  tuple<vc<Ts>...> tv;
  apply([&](auto &...v)
        { ((v.resize(n)), ...); }, tv);
  for (size_t i = 0; i < n; i++)
    internal::vt_to_tv_impl(tv, vt[i], make_index_sequence<tuple_size_v<decltype(tv)>>{}, i);
  return tv;
}
template <class... Ts>
auto top(const tuple<vc<Ts>...> &tv)
{
  size_t n = get<0>(tv).size();
  apply([&](auto &...v)
        { ((assert(v.size() == n)), ...); }, tv);
  vc<tuple<Ts...>> vt(n);
  for (size_t i = 0; i < n; i++)
    vt[i] = internal::tv_to_vt_impl(tv, index_sequence_for<Ts...>{}, i);
  return vt;
}
mt19937_64 mt;
template <class T>
struct Binomial
{
private:
  static decltype(T::mod()) mod;
  static vc<T> fac_, finv_, inv_;
public:
  static void reserve(int n)
  {
    if (mod != T::mod())
    {
      mod = T::mod();
      fac_ = {1, 1}, finv_ = {1, 1}, inv_ = {0, 1};
    }
    int i = fac_.size();
    chmin(n, T::mod() - 1);
    if (n < i)
      return;
    fac_.resize(n + 1), finv_.resize(n + 1), inv_.resize(n + 1);
    for (; i <= n; i++)
    {
      fac_[i] = fac_[i - 1] * T::raw(i);
      inv_[i] = -inv_[T::mod() % i] * T::raw(T::mod() / i);
      finv_[i] = finv_[i - 1] * inv_[i];
    }
  }
  static T fac(int n)
  {
    assert(n >= 0);
    if (n >= T::mod())
      return 0;
    reserve(n);
    return fac_[n];
  }
  static T inv(T n)
  {
    assert(n != 0);
    reserve(n.val());
    return inv_[n.val()];
  }
};
template <class T> decltype(T::mod()) Binomial<T>::mod{};
template <class T> vc<T> Binomial<T>::fac_{};
template <class T> vc<T> Binomial<T>::finv_{};
template <class T> vc<T> Binomial<T>::inv_{};
namespace internal
{
constexpr ll powmod32_constexpr(ll x, ll n, int m)
{
  if (m == 1)
    return 0;
  uint _m = (uint)m;
  ull r = 1;
  ull y = safemod(x, m);
  while (n)
  {
    if (n & 1)
      r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}
constexpr bool isprime32_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;
  constexpr ll bases[3] = {2, 7, 61};
  for (ll a : bases)
  {
    ll t = d;
    ll y = powmod32_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y = y * y % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}
template <int n>
constexpr bool isprime32 = isprime32_constexpr(n);
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));
  }
};
}
namespace internal
{
#define REF static_cast<mint &>(*this)
#define CREF static_cast<const mint &>(*this)
#define VAL *static_cast<const mint *>(this)
template <class mint>
struct modint_base
{
  mint &operator+=(const mint &rhs)
  {
    mint &self = REF;
    self._v += rhs._v;
    if (self._v >= self.umod())
      self._v -= self.umod();
    return self;
  }
  mint &operator-=(const mint &rhs)
  {
    mint &self = REF;
    self._v -= rhs._v;
    if (self._v >= self.umod())
      self._v += self.umod();
    return self;
  }
  mint &operator/=(const mint &rhs)
  {
    mint &self = REF;
    return self = self * rhs.inv();
  }
  mint &operator++()
  {
    mint &self = REF;
    self._v++;
    if (self._v == self.umod())
      self._v = 0;
    return self;
  }
  mint &operator--()
  {
    mint &self = REF;
    if (self._v == 0)
      self._v = self.umod();
    self._v--;
    return self;
  }
  mint operator++(int)
  {
    mint res = VAL;
    ++REF;
    return res;
  }
  mint operator--(int)
  {
    mint res = VAL;
    --REF;
    return res;
  }
  mint operator+() const { return VAL; }
  mint operator-() const { return mint() - VAL; }
  mint pow(ll n) const
  {
    assert(n >= 0);
    mint x = VAL, r = 1;
    while (n)
    {
      if (n & 1)
        r *= x;
      x *= x;
      n >>= 1;
    }
    return r;
  }
  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 mint(lhs).eq(rhs); }
  friend bool operator!=(const mint &lhs, const mint &rhs)
  { return mint(lhs).neq(rhs); }
private:
  bool eq(const mint &rhs) { return REF._v == rhs._v; }
  bool neq(const mint &rhs) { return REF._v != rhs._v; }
};
}
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
void rd1(T &x)
{
  ll a;
  fastio::rd1(a);
  x = a;
}
template <typename T, std::enable_if_t<std::is_base_of_v<internal::modint_base<T>, T>, int> = 0>
void wt1(const T &x) { fastio::wt1(x.val()); }
template <class T = ll>
constexpr tuple<T, T, T> extgcd(const T &a, const 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};
}
template <int m>
struct static_modint : internal::modint_base<static_modint<m>>
{
  using mint = static_modint;
private:
  friend struct internal::modint_base<static_modint<m>>;
  uint _v;
  static constexpr uint umod() { return m; }
  static constexpr bool prime = internal::isprime32<m>;
public:
  static constexpr int mod() { return m; }
  static mint raw(int v)
  {
    mint x;
    x._v = v;
    return x;
  }
  static_modint() : _v(0) {}
  template <class T>
  static_modint(T v)
  {
    if constexpr (is_signed_v<T>)
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (uint)x;
    }
    else if constexpr (is_unsigned_v<T>)
    {
      _v = (uint)(v % umod());
    }
    else
    {
      static_assert(is_signed_v<T> || is_unsigned_v<T>, "Unsupported Type");
    }
  }
  int val() const { return (int)_v; }
  mint& operator*=(const mint &rhs)
  {
    ull z = _v;
    z *= rhs._v;
    _v = (uint)(z % umod());
    return *this;
  }
  mint inv() const
  {
    if (prime)
    {
      assert(_v != 0);
      return CREF.pow(umod() - 2);
    }
    else
    {
      auto [g, x, y] = extgcd<int>(_v, m);
      assert(g == 1);
      return x;
    }
  }
};
template <int id>
struct dynamic_modint : internal::modint_base<dynamic_modint<id>>
{
  using mint = dynamic_modint;
private:
  friend struct internal::modint_base<dynamic_modint<id>>;
  uint _v;
  static internal::barrett32 bt;
  static uint umod() { return bt.umod(); }
public:
  static int mod() { return (int)(bt.umod()); }
  static mint raw(int v)
  {
    mint x;
    x._v = v;
    return x;
  }
  dynamic_modint() : _v(0) {}
  template <class T>
  dynamic_modint(T v)
  {
    if constexpr (is_signed_v<T>)
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (uint)x;
    }
    else if constexpr (is_unsigned_v<T>)
    {
      _v = (uint)(v % umod());
    }
    else
    {
      static_assert(is_signed_v<T> || is_unsigned_v<T>, "Unsupported Type");
    }
  }
  int val() const { return (int)_v; }
  mint& operator*=(const mint &rhs)
  {
    _v = bt.mul(_v, rhs._v);
    return *this;
  }
  mint inv() const
  {
    auto [g, x, y] = extgcd<int>(_v, mod());
    assert(g == 1);
    return x;
  }
};
template <int id>
internal::barrett32 dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
template <class T>
struct is_static_modint : false_type {};
template <int m>
struct is_static_modint<static_modint<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_modint<id>> : true_type {};
template <class T>
inline constexpr bool is_dynamic_modint_v = is_dynamic_modint<T>::value;
template <class mint, class T = ll, class U1, class U2, size_t n>
constexpr pair<mint, mint> crt_mod_constexpr(const array<U1, n> &rs, const array<U2, n> &ms)
{
  assert(rs.size() == ms.size());
  mint r = 0, m = 1;
  array<T, n> rr{}, mm;
  fill(ALL(mm), 1);
  repi(i, n)
  {
    assert(ms[i] >= U2(1));
    assert(U1(0) <= rs[i] && U2(rs[i]) < ms[i]);
    auto [g, im, _] = extgcd<T>(mm[i], ms[i]);
    assert(g == 1);
    T t = safemod((rs[i] - rr[i]) * im, ms[i]);
    r += t * m, m *= ms[i];
    repi(j, i + 1, n)
    {
      rr[j] += t * mm[j] % ms[j];
      if (rr[j] >= ms[j])
        rr[j] -= ms[j];
      mm[j] *= ms[i], mm[j] %= ms[j];
    }
  }
  return {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 (powmod32_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, rank2 + 1> root;  // root[i]^(2^i) == 1
  std::array<mint, rank2 + 1> iroot; // root[i] * iroot[i] == 1
  std::array<mint, std::max(0, rank2 - 2 + 1)> rate2;
  std::array<mint, std::max(0, rank2 - 2 + 1)> irate2;
  std::array<mint, std::max(0, rank2 - 3 + 1)> rate3;
  std::array<mint, std::max(0, 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];
      }
    }
  }
};
}  // namespace internal
template <class mint>
bool ntt_ok(int n)
{
  if constexpr (is_static_modint_v<mint>)
  {
    if constexpr (!internal::isprime32<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_modint<id>> &) {}
template <int id>
void intt(vc<dynamic_modint<id>> &) {}
template <int mod>
void ntt(vc<static_modint<mod>> &a)
{
  using mint = static_modint<mod>;
  int n = int(a.size());
  int h = countr_zero((unsigned int)n);
  static const internal::fft_info<mint> info;
  int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
  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 <int mod>
void intt(vc<static_modint<mod>> &a)
{
  using mint = static_modint<mod>;
  int n = int(a.size());
  int h = countr_zero((unsigned int)n);
  static const internal::fft_info<mint> info;
  int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
  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_modint<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 <class mint, int... ms, class T>
vc<mint> convolution_crt_mod(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<mint> c(n + m - 1);
  repi(i, n + m - 1) c[i] = crt_mod_constexpr<mint>(cs[i], ms_arr).first;
  return c;
}
}  // namespace internal
template <class mint, typename = std::enable_if_t<!std::is_integral<mint>::value>>
vc<mint> convolution(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);
  if (n == 0 || m == 0)
    return {};
  if (ntt_ok<mint>(n + m - 1))
  {
    if (min(cnta, cntb) <= 60)
      return internal::convolution_naive(a, b);
    return internal::convolution_ntt(a, b);
  }
  else
  {
    if (min(cnta, cntb) <= 300)
      return internal::convolution_naive(a, b);
    assert(ntt_ok<static_modint<469762049>>(n + m - 1) && "|a| + |b| - 1 <= 2^26");
    vc<ll> a_(n), b_(m);
    repi(i, n) a_[i] = a[i].val();
    repi(j, m) b_[j] = b[j].val();
    return internal::convolution_crt_mod<mint, 469762049, 1811939329, 2013265921>(a_, b_);
  }
}
template <int mod = 998244353, class T, typename = enable_if_t<is_integral<T>::value>>
vc<T> convolution(const vc<T> &a, const vc<T> &b)
{
  using mint = static_modint<mod>;
  auto c = convolution(vc<mint>(ALL(a)), vc<mint>(ALL(b)));
  vc<T> c_(c.size());
  repi(i, c.size()) c_[i] = c[i].val();
  return c_;
}
namespace internal
{
constexpr ll powmod64_constexpr(ll x, ll n, ll m)
{
  if (m == 1)
    return 0;
  ull _m = (ull)m;
  ull r = 1;
  ull y = safemod(x, m);
  while (n)
  {
    u128 y128(y);
    if (n & 1)
      r = (y128 * r) % _m;
    y = (y128 * y) % _m;
    n >>= 1;
  }
  return r;
}
constexpr bool isprime64_constexpr(ll n)
{
  if (n <= INT_MAX)
    return isprime32_constexpr(n);
  if (n % 2 == 0)
    return false;
  ll d = n - 1;
  while (d % 2 == 0)
    d /= 2;
  constexpr ll bases[7] = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};
  for (ll a : bases)
  {
    ll t = d;
    ll y = powmod64_constexpr(a, t, n);
    while (t != n - 1 && y != 1 && y != n - 1)
    {
      y = (u128(y) * y) % n;
      t <<= 1;
    }
    if (y != n - 1 && t % 2 == 0)
      return false;
  }
  return true;
}
template <ll n>
constexpr bool isprime64 = isprime64_constexpr(n);
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;  // m == 2^b * mx, mx is odd
    imx = inv64(mx);
    d = powmod64_constexpr((mx + 1) / 2, b, mx);  // 2^{-b} mod mx
    u128 sq = -u128(mx) % mx;  // 2^128 mod mx
    q = (1 + (((sq - 1) * d) << b)) % m;
  }
  ull umod() const { return m; }
  ull reduce(u128 x) const
  {
    ull p = x & MASK(b);  // x mod 2^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 <ll m>
struct static_modint64 : internal::modint_base<static_modint64<m>>
{
  using mint = static_modint64;
private:
  friend struct internal::modint_base<static_modint64<m>>;
  ull _v;
  static constexpr ull umod() { return m; }
  static constexpr bool prime = internal::isprime64<m>;
public:
  static constexpr ll mod() { return m; }
  static mint raw(ll v)
  {
    mint x;
    x._v = v;
    return x;
  }
  static_modint64() : _v(0) {}
  template <class T>
  static_modint64(T v)
  {
    if constexpr (is_unsigned_v<T>)
    {
      _v = (ull)(v % umod());
    }
    else
    {
      ll x = (ll)(v % (ll)(umod()));
      if (x < 0)
        x += umod();
      _v = (ull)x;
    }
  }
  ll val() const { return (ll)_v; }
  mint& operator*=(const mint &rhs)
  {
    u128 z = _v;
    z *= rhs._v;
    _v = (ull)(z % umod());
    return *this;
  }
  mint inv() const
  {
    if (prime)
    {
      assert(_v != 0);
      return CREF.pow(umod() - 2);
    }
    else
    {
      auto [g, x, y] = extgcd<ll>(_v, m);
      assert(g == 1);
      return x;
    }
  }
};
template <int id>
struct dynamic_modint64_odd : internal::modint_base<dynamic_modint64_odd<id>>
{
  using mint = dynamic_modint64_odd;
private:
  friend struct internal::modint_base<dynamic_modint64_odd<id>>;
  ull _v; // montgomery expression
  static internal::montgomery64odd mg;
  static ull umod() { return mg.umod(); }
public:
  static ll mod() { return (ll)(mg.umod()); }
  dynamic_modint64_odd() : _v(0) {}
  dynamic_modint64_odd(i128 v)
  { _v = mg.inv_reduce(v); }
  ll val() const { return (ll)mg.reduce(_v); }
  mint& operator*=(const mint &rhs)
  {
    _v = mg.reduce(u128(_v) * rhs._v);
    return *this;
  }
  mint inv() const
  {
    auto [g, x, y] = extgcd<ll>(val(), mod());
    assert(g == 1);
    return x;
  }
};
template <int id>
internal::montgomery64odd dynamic_modint64_odd<id>::mg((1LL << 61) - 1);
template <int id>
struct dynamic_modint64 : internal::modint_base<dynamic_modint64<id>>
{
  using mint = dynamic_modint64;
private:
  friend struct internal::modint_base<dynamic_modint64<id>>;
  ull _v; // montgomery expression
  static internal::montgomery64 mg;
  static ull umod() { return mg.umod(); }
public:
  static ll mod() { return (ll)(mg.umod()); }
  dynamic_modint64() : _v(0) {}
  dynamic_modint64(i128 v)
  { _v = mg.inv_reduce(v); }
  ll val() const { return (ll)mg.reduce(_v); }
  mint& operator*=(const mint &rhs)
  {
    _v = mg.reduce(u128(_v) * rhs._v);
    return *this;
  }
  mint inv() const
  {
    auto [g, x, y] = extgcd<ll>(val(), mod());
    assert(g == 1);
    return x;
  }
};
template <int id>
internal::montgomery64 dynamic_modint64<id>::mg((1LL << 61) - 1);
namespace internal
{
}; // namespace internal
template <class mint>
struct FormalPowerSeries : vc<mint>
{
  using F = FormalPowerSeries;
  using vc<mint>::vc;
  using vc<mint>::operator=;
  using vc<mint>::size;
  using vc<mint>::empty;
  using vc<mint>::back;
  using vc<mint>::pop_back;
  using vc<mint>::begin;
  using vc<mint>::resize;
  using vc<mint>::front;
  FormalPowerSeries(const vc<mint> &f) : vc<mint>(f) {}
  int sz() const { return size(); }
  void shrink()
  {
    while (!empty() && back() == 0)
      pop_back();
  }
  mint get(int i) const { return 0 <= i && i < sz() ? (*this)[i] : 0; }
  F pre(int len) const
  {
    assert(len >= 0);
    return F(begin(), begin() + min(sz(), len));
  }
  F rev(int d = -1) const
  {
    F res(*this);
    if (d >= 0)
      res.resize(d);
    reverse(ALL(res));
    return res;
  }
  int cnt_nz() const { return count_if(ALL(*this), LMD(x, x != 0)); }
  tuple<bool, int, mint> nz_front() const
  {
    repi(i, sz()) if ((*this)[i] != 0) return {true, i, (*this)[i]};
    return {false, -1, 0};
  }
  vc<pair<int, mint>> nz() const
  {
    vc<pair<int, mint>> res;
    repi(i, sz()) if ((*this)[i] != 0) res.eb(i, (*this)[i]);
    return res;
  }
  F operator-() const
  {
    F res(*this);
    fem(a : res) a = -a;
    return res;
  }
  F &operator*=(const mint &k)
  {
    fem(a : *this) a *= k;
    return *this;
  }
  F operator*(const mint &k) const { return F(*this) *= k; }
  friend F operator*(const mint &k, const F &f) { return f * k; }
  F &operator/=(const mint &k)
  {
    *this *= k.inv();
    return *this;
  }
  F operator/(const mint &k) const { return F(*this) /= k; }
  F &operator+=(const F &g)
  {
    const int n = size(), m = g.size();
    resize(max(n, m));
    repi(i, m)(*this)[i] += g[i];
    return *this;
  }
  F operator+(const F &g) const { return F(*this) += g; }
  F &operator-=(const F &g)
  {
    const int n = size(), m = g.size();
    resize(max(n, m));
    repi(i, m)(*this)[i] -= g[i];
    return *this;
  }
  F operator-(const F &g) const { return F(*this) -= g; }
  F &operator*=(const F &g) { return *this = *this * g; }
  F operator*(const F &g) const { return convolution(*this, g); }
  F div_sparse_destructive(const F &g, int d = -1)
  {
    assert(g.get(0) != 0);
    if (d < 0)
      d = sz();
    mint iv = g.front().inv();
    auto gnz = g.nz();
    resize(d);
    repi(i, d)
    {
      fec([j, b] : gnz)
      {
        if (j == 0)
          continue;
        if (j > i)
          break;
        (*this)[i] -= (*this)[i - j] * b;
      }
      (*this)[i] *= iv;
    }
    return pre(d);
  }
  F div_sparse(const F &g, int d = -1) const { return F(*this).div_sparse_destructive(g, d); }
  F inv(int d = -1) const
  {
    assert(get(0) != 0);
    if (d < 0)
      d = sz();
    if (cnt_nz() <= 200)
      return F{1}.div_sparse(*this, d);
    F f, g2, g{front().inv()};
    for (int m = 1; m < d; m *= 2)
    {
      if (ntt_ok<mint>(2 * m))
      {
        f = pre(2 * m), g2 = F(g);
        f.resize(2 * m), ntt(f);
        g2.resize(2 * m), ntt(g2);
        repi(i, 2 * m) f[i] *= g2[i];
        intt(f);
        f >>= m;
        f.resize(2 * m), ntt(f);
        repi(i, 2 * m) f[i] *= g2[i];
        intt(f);
        mint iz = mint(2 * m).inv();
        iz *= -iz;
        repi(i, m) f[i] *= iz;
        g.insert(g.end(), f.begin(), f.begin() + m);
      }
      else
        g = (g * mint(2) - g * g * pre(2 * m)).pre(2 * m);
    }
    return g.pre(d);
  }
  F &operator/=(const F &g)
  {
    if (cnt_nz() <= 200)
    {
      div_sparse_destructive(g);
      return *this;
    }
    *this *= g.inv();
    return *this;
  }
  F operator/(const F &g) const { return F(*this) /= g; }
  F div_poly(const F &g) const
  {
    const int n = sz() - g.sz() + 1;
    if (n <= 0)
      return {};
    return (rev().pre(n) * g.rev().inv(n)).pre(n).rev();
  }
  pair<F, F> divmod(const F &g) const
  {
    F q = div_poly(g);
    F r = *this - q * g;
    r.shrink();
    return {q, r};
  }
  F operator%(const F &g) const { return divmod(g).second; }
  F &operator%=(const F &g) { return *this = *this % g; }
  F circular_mod(int n) const
  {
    F res(n);
    repi(i, sz()) res[i % n] += (*this)[i];
    return res;
  }
  F operator<<(int k) const
  {
    F res(sz() + k);
    repi(i, sz()) res[i + k] = (*this)[i];
    return res;
  }
  F operator>>(int k) const
  {
    F res(max(0, sz() - k));
    repi(i, sz() - k) res[i] = (*this)[i + k];
    return res;
  }
  F &operator<<=(int k) { return *this = *this << k; }
  F &operator>>=(int k) { return *this = *this >> k; }
  F diff() const
  {
    F res(max(0, sz() - 1));
    repi(i, 1, size()) res[i - 1] = (*this)[i] * i;
    return res;
  }
  F integ() const
  {
    F res(sz() + 1);
    repi(i, size()) res[i + 1] = (*this)[i] * Binomial<mint>::inv(i + 1);
    return res;
  }
  F log(int d = -1) const
  {
    assert(get(0) == 1);
    if (d < 0)
      d = sz();
    F f = pre(d);
    return (f.diff() / f).pre(d - 1).integ();
  }
  static F diff_eq(const F &a, const F &b, int d)
  {
    assert(a.get(0) == 1);
    assert(d >= 0);
    if (d == 0)
      return {};
    F f(d);
    f[0] = 1;
    auto anz = a.nz(), bnz = b.nz();
    repi(k, d - 1)
    {
      fec([i, ai] : anz)
      {
        if (0 <= k - i + 1)
          f[k + 1] -= ai * (k - i + 1) * f[k - i + 1];
      }
      fec([j, bj] : bnz)
      {
        if (0 <= k - j && k - j < k + 1)
          f[k + 1] -= bj * f[k - j];
      }
      f[k + 1] *= Binomial<mint>::inv(k + 1);
    }
    return f;
  }
  F exp_sparse(int d = -1) const
  {
    assert(get(0) == 0);
    if (d < 0)
      d = sz();
    return diff_eq(F{1}, -diff(), d);
  }
  F pow_sparse(ll k, int d = -1) const
  {
    if (d < 0)
      d = sz();
    auto [exi, d0, a0] = nz_front();
    if (!exi)
    {
      F res(d);
      if (k == 0 && d > 0)
        res[0] = 1;
      return res;
    }
    mint ia0 = a0.inv();
    F f = ((*this) >> d0) * ia0;
    if (k >= 0)
    {
      F g = diff_eq(f, -k * f.diff(), d - mul_limited(d0, k, d));
      F h = (g * a0.pow(k)) << mul_limited(d0, k, d);
      return h.pre(d);
    }
    else
    {
      F g = diff_eq(f, -k * f.diff(), d + (d0 * (-k)));
      F h = (g * ia0.pow(-k)) >> (d0 * (-k));
      return h.pre(d);
    }
  }
  F exp(int d = -1) const
  {
    assert(get(0) == 0);
    if (d < 0)
      d = sz();
    if (ntt_ok<mint>(2 * d))
    {
      if (cnt_nz() <= 320)
        return exp_sparse(d);
      F f{1}, g{1};
      F f2, g2, f3, q, s, h, u;
      g2 = {0};
      for (int m = 1; m < d; m *= 2)
      {
        mint im = mint(m).inv(), i2m = mint(2 * m).inv();
        f2 = f, f2.resize(2 * m), ntt(f2);
        f3 = f, ntt(f3);
        repi(i, m) f3.at(i) *= g2.at(i);
        intt(f3);
        f3 >>= m / 2;
        f3.resize(m), ntt(f3);
        repi(i, m) f3.at(i) *= g2.at(i);
        intt(f3);
        repi(i, m / 2) f3.at(i) *= -im * im;
        g.insert(g.end(), f3.begin(), f3.begin() + m / 2);
        g2 = g, g2.resize(2 * m), ntt(g2);
        q = diff(), q.resize(2 * m), fill(q.begin() + m - 1, q.end(), 0);
        ntt(q);
        repi(i, 2 * m) q.at(i) *= f2.at(i);
        intt(q);
        q = q.circular_mod(m);
        repi(i, m) q.at(i) *= i2m;
        q.resize(m + 1);
        s = ((f.diff() - q) << 1).circular_mod(m);
        s.resize(2 * m), ntt(s);
        repi(i, 2 * m) s.at(i) *= g2.at(i);
        intt(s);
        repi(i, m) s.at(i) *= i2m;
        s.resize(m);
        h = *this, h.resize(2 * m), s.resize(2 * m);
        u = (h - (s << (m - 1)).integ()) >> m;
        ntt(u);
        repi(i, 2 * m) u.at(i) *= f2.at(i);
        intt(u);
        repi(i, m) u.at(i) *= i2m;
        u.resize(m);
        f.insert(f.end(), u.begin(), u.end());
      }
      return f.pre(d);
    }
    else
    {
      if (cnt_nz() <= 3000)
        return exp_sparse(d);
      F f{1};
      for (int m = 1; m < d; m *= 2)
      {
        f = (f * (pre(2 * m) + F{1} - f.log(2 * m))).pre(2 * m);
      }
      return f.pre(d);
    }
  }
  F pow(ll k, int d = -1) const
  {
    if (ntt_ok<mint>(2 * d))
    {
      if (cnt_nz() <= 100)
        return pow_sparse(k, d);
    }
    else
    {
      if (cnt_nz() <= 1300)
        return pow_sparse(k, d);
    }
    if (d < 0)
      d = sz();
    if (k == 0)
    {
      F res(d);
      res[0] = 1;
      return res;
    }
    repi(i, sz())
    {
      if ((*this)[i] != 0)
      {
        mint iv = (*this)[i].inv();
        F res = (((*this * iv) >> i).log(d) * mint(k)).exp(d);
        res *= (*this)[i].pow(k);
        res = (res << (i * k)).pre(d);
        if (res.sz() < d)
          res.resize(d);
        return res;
      }
      if (mul_limited(i + 1, k, d) >= d)
        return F(d);
    }
    return F(d);
  }
};
using mint = modint998244353;
using bi = Binomial<mint>;
using fps = FormalPowerSeries<mint>;
template <class mint>
pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>> rational_plus
(
  const pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>> &f,
  const pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>> &g
)
{
  using F = FormalPowerSeries<mint>;
  cauto &[p_, q_] = f;
  cauto &[r_, s_] = g;
  if (min({p_.cnt_nz(), q_.cnt_nz(), r_.cnt_nz(), s_.cnt_nz()}) <= 60)
    return {p_ * s_ + q_ * r_, q_ * s_};
  F p = p_, q = q_, r = r_, s = s_;
  const int k = p.size(), l = q.size(), m = r.size(), n = s.size();
  const int z = bit_ceil(max({k + n - 1, l + m - 1, l + n - 1}));
  p.resize(z), q.resize(z), r.resize(z), s.resize(z);
  ntt(p), ntt(q), ntt(r), ntt(s);
  F a(z), b(z), c(z);
  repi(i, z) a[i] = p[i] * s[i], b[i] = q[i] * r[i], c[i] = q[i] * s[i];
  intt(a), intt(b), intt(c);
  mint iz = mint(z).inv();
  repi(i, z) a[i] *= iz, b[i] *= iz, c[i] *= iz;
  return {a.pre(k + n - 1) + b.pre(l + m - 1), c.pre(l + n - 1)};
}
template <class mint>
pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>> rational_sum
(const vc<pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>>> &fs, int d = -1)
{
  using F = FormalPowerSeries<mint>;
  using R = pair<F, F>;
  auto dc = [&](auto dc, int l, int r) -> R
  {
    if (r - l == 0)
      return {{1}, {1}};
    if (r - l == 1)
      return fs[l];
    const int m = (l + r) / 2;
    R res = rational_plus(dc(dc, l, m), dc(dc, m, r));
    if (d < 0)
      return res;
    else
      return {res.first.pre(d), res.second.pre(d)};
  };
  return dc(dc, 0, fs.size());
}
void init() {}
void main2()
{
  LL(N, S);
  VEC(mint, N, P);
  fem(p : P) p /= S;
  vc<pair<fps, fps>> fs(N);
  rep(i, N) fs.at(i) = {{0, P.at(i) * P.at(i)}, {1, P.at(i)}};
  auto f = rational_sum(fs);
  dump(fs, f);
  mint ans = 0;
  rep(k, N + 1)
  {
    ans += (k + 1) * bi::fac(k) * f.first.at(k);
    dump(k, f.first.at(k), bi::fac(k) * f.first.at(k));
  }
  PRINT(ans);
}
void test()
{
}
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);
    test();
    init();
    CERR("\n[SINGLE_TESTCASE]\n\n", "36");
    main2();
  }
};
Main<init, main2, test> main_dummy;
}
int main() {}