/**
 * date   : 2024-03-01 22:46:32
 * author : Nyaan
 */

#define NDEBUG

using namespace std;

// intrinstic
#include <immintrin.h>

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

// utility

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

}  // namespace Nyaan


// bit operation

namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
  return _mm_popcnt_u64(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
  return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
  if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
}  // namespace Nyaan


// inout

namespace Nyaan {

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

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

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

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

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

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

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

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

}  // namespace Nyaan


// debug


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

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


// macro

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


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


//



using namespace std;




using namespace std;

// コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」
// を入れると倍速くらいになる
// mod を超えて前計算して 0 割りを踏むバグは対策済み
template <typename T>
struct Binomial {
  vector<T> f, g, h;
  Binomial(int MAX = 0) {
    assert(T::get_mod() != 0 && "Binomial<mint>()");
    f.resize(1, T{1});
    g.resize(1, T{1});
    h.resize(1, T{1});
    if (MAX > 0) extend(MAX + 1);
  }

  void extend(int m = -1) {
    int n = f.size();
    if (m == -1) m = n * 2;
    m = min<int>(m, T::get_mod());
    if (n >= m) return;
    f.resize(m);
    g.resize(m);
    h.resize(m);
    for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
    g[m - 1] = f[m - 1].inverse();
    h[m - 1] = g[m - 1] * f[m - 2];
    for (int i = m - 2; i >= n; i--) {
      g[i] = g[i + 1] * T(i + 1);
      h[i] = g[i] * f[i - 1];
    }
  }

  T fac(int i) {
    if (i < 0) return T(0);
    while (i >= (int)f.size()) extend();
    return f[i];
  }

  T finv(int i) {
    if (i < 0) return T(0);
    while (i >= (int)g.size()) extend();
    return g[i];
  }

  T inv(int i) {
    if (i < 0) return -inv(-i);
    while (i >= (int)h.size()) extend();
    return h[i];
  }

  T C(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r) * finv(r);
  }

  inline T operator()(int n, int r) { return C(n, r); }

  template <typename I>
  T multinomial(const vector<I>& r) {
    static_assert(is_integral<I>::value == true);
    int n = 0;
    for (auto& x : r) {
      if (x < 0) return T(0);
      n += x;
    }
    T res = fac(n);
    for (auto& x : r) res *= finv(x);
    return res;
  }

  template <typename I>
  T operator()(const vector<I>& r) {
    return multinomial(r);
  }

  T C_naive(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    T ret = T(1);
    r = min(r, n - r);
    for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
    return ret;
  }

  T P(int n, int r) {
    if (n < 0 || n < r || r < 0) return T(0);
    return fac(n) * finv(n - r);
  }

  // [x^r] 1 / (1-x)^n
  T H(int n, int r) {
    if (n < 0 || r < 0) return T(0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};


template <typename mint>
struct FormalPowerSeries : vector<mint> {
  using vector<mint>::vector;
  using FPS = FormalPowerSeries;

  FPS &operator+=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
    return *this;
  }

  FPS &operator+=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }

  FPS &operator-=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
    return *this;
  }

  FPS &operator-=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }

  FPS &operator*=(const mint &v) {
    for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
    return *this;
  }

  FPS &operator/=(const FPS &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    if ((int)r.size() <= 64) {
      FPS f(*this), g(r);
      g.shrink();
      mint coeff = g.back().inverse();
      for (auto &x : g) x *= coeff;
      int deg = (int)f.size() - (int)g.size() + 1;
      int gs = g.size();
      FPS quo(deg);
      for (int i = deg - 1; i >= 0; i--) {
        quo[i] = f[i + gs - 1];
        for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
      }
      *this = quo * coeff;
      this->resize(n, mint(0));
      return *this;
    }
    return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
  }

  FPS &operator%=(const FPS &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }

  FPS operator+(const FPS &r) const { return FPS(*this) += r; }
  FPS operator+(const mint &v) const { return FPS(*this) += v; }
  FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
  FPS operator-(const mint &v) const { return FPS(*this) -= v; }
  FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
  FPS operator*(const mint &v) const { return FPS(*this) *= v; }
  FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
  FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
  FPS operator-() const {
    FPS ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }

  void shrink() {
    while (this->size() && this->back() == mint(0)) this->pop_back();
  }

  FPS rev() const {
    FPS ret(*this);
    reverse(begin(ret), end(ret));
    return ret;
  }

  FPS dot(FPS r) const {
    FPS ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }

  // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
  FPS pre(int sz) const {
    FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz));
    if ((int)ret.size() < sz) ret.resize(sz);
    return ret;
  }

  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }

  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret.insert(ret.begin(), sz, mint(0));
    return ret;
  }

  FPS diff() const {
    const int n = (int)this->size();
    FPS ret(max(0, n - 1));
    mint one(1), coeff(1);
    for (int i = 1; i < n; i++) {
      ret[i - 1] = (*this)[i] * coeff;
      coeff += one;
    }
    return ret;
  }

  FPS integral() const {
    const int n = (int)this->size();
    FPS ret(n + 1);
    ret[0] = mint(0);
    if (n > 0) ret[1] = mint(1);
    auto mod = mint::get_mod();
    for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
    for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
    return ret;
  }

  mint eval(mint x) const {
    mint r = 0, w = 1;
    for (auto &v : *this) r += w * v, w *= x;
    return r;
  }

  FPS log(int deg = -1) const {
    assert(!(*this).empty() && (*this)[0] == mint(1));
    if (deg == -1) deg = (int)this->size();
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }

  FPS pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if (k == 0) {
      FPS ret(deg);
      if (deg) ret[0] = 1;
      return ret;
    }
    for (int i = 0; i < n; i++) {
      if ((*this)[i] != mint(0)) {
        mint rev = mint(1) / (*this)[i];
        FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
        ret *= (*this)[i].pow(k);
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
      if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
    }
    return FPS(deg, mint(0));
  }

  static void *ntt_ptr;
  static void set_fft();
  FPS &operator*=(const FPS &r);
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_pr();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;

/**
 * @brief 多項式/形式的冪級数ライブラリ
 * @docs docs/fps/formal-power-series.md
 */




// find Q(P(x)) mod x ^ min(deg(P), deg(Q))
template <typename mint>
FormalPowerSeries<mint> Composition(FormalPowerSeries<mint> P,
                                    FormalPowerSeries<mint> Q,
                                    Binomial<mint>& C, int deg = -1) {
  using fps = FormalPowerSeries<mint>;
  int N = (deg == -1) ? min(P.size(), Q.size()) : deg;
  if (N == 0) return fps{};
  P.shrink();
  if (P.size() == 0) {
    fps R(N, mint(0));
    R[0] = Q.empty() ? mint(0) : Q[0];
    return R;
  }
  if (N == 1) return fps{Q.eval(P[0])};

  P.resize(N, mint(0));
  Q.resize(N, mint(0));
  int M = max<int>(1, sqrt(N / log2(N)));
  int L = (N + M - 1) / M;
  fps Pm = fps{begin(P), begin(P) + M};
  fps Pr = fps{begin(P) + M, end(P)};

  int J = 31 - __builtin_clz(N - 1) + 1;
  vector<fps> pms(J);
  pms[0] = Pm;
  for (int i = 1; i < J; i++) pms[i] = (pms[i - 1] * pms[i - 1]).pre(N);

  auto comp = [&](auto rec, int left, int j) -> fps {
    if (j == 1) {
      mint Q1 = left + 0 < (int)Q.size() ? Q[left + 0] : mint(0);
      mint Q2 = left + 1 < (int)Q.size() ? Q[left + 1] : mint(0);
      return (pms[0].pre(N) * Q2 + Q1).pre(N);
    }
    if (N <= left) return fps{};
    fps Q1 = rec(rec, left, j - 1);
    fps Q2 = rec(rec, left + (1 << (j - 1)), j - 1);
    return (Q1 + pms[j - 1].pre(N) * Q2).pre(N);
  };

  fps QPm = comp(comp, 0, J);
  fps R = QPm;
  fps pw_Pr{mint(1)};
  fps dPm = Pm.diff();
  dPm.shrink();
  // if dPm[0] == 0, dPm.inv() is undefined
  int deg_dPm = 0;
  while (deg_dPm != (int)dPm.size() && dPm[deg_dPm] == mint(0)) deg_dPm++;
  fps idPm = dPm.empty() ? fps{} : (dPm >> deg_dPm).inv(N);

  for (int l = 1, d = M; l <= L && d < N; l++, d += M) {
    pw_Pr = (pw_Pr * Pr).pre(N - d);
    if (dPm.empty()) {
      R += (pw_Pr * Q[l]) << d;
    } else {
      idPm.resize(N - d);
      QPm = ((QPm.diff() >> deg_dPm) * idPm).pre(N - d);
      R += ((QPm * pw_Pr).pre(N - d) * C.finv(l)) << d;
    };
  }
  R.resize(N, mint(0));
  return R;
}

/**
 * @brief 関数の合成( $\mathrm{O}\left((N \log N)^{\frac{3}{2}}\right)$ )
 * @docs docs/fps/fps-composition.md
 */


// f を入力として, f(g(x)) = x を満たす g(x) mod x^{deg} を返す
template <typename fps>
fps compositional_inverse(fps f, int deg = -1) {
  using mint = typename fps::value_type;
  if (deg == -1) deg = f.size();
  assert((int)f.size() >= 2 && f[0] == 0 && f[1] != 0);
  fps g{0, f[1].inverse()};
  Binomial<mint> binom;
  for (int d = 2; d < deg; d *= 2) {
    fps f2{begin(f), begin(f) + min(2 * d + 1, (int)f.size())};
    fps fg = Composition<mint>(g, f2, binom, 2 * d + 1);
    fps fdg = (fg.diff() * g.diff().inv(2 * d)).pre(2 * d);
    assert(fdg[0] != 0);
    g = (g - (fg - fps{0, 1}) * fdg.inv(2 * d)).pre(2 * d);
  }
  return {begin(g), begin(g) + deg};
}

// f(g(x)) = x を満たす g(x) mod x^{deg} を返す
// calc_f(g, d) は f(g(x)) mod x^d を計算する関数
template <typename fps>
fps compositional_inverse(function<fps(fps, int)> calc_f, int deg) {
  fps g = calc_f(fps{0, 1}, 2);
  assert(g[0] == 0 && g[1] != 0);
  g[1] = g[1].inverse();
  for (int d = 2; d < deg; d *= 2) {
    fps fg = calc_f(g, 2 * d + 1);
    fps fdg = (fg.diff() * g.diff().inv(2 * d)).pre(2 * d);
    g = (g - (fg - fps{0, 1}) * fdg.inv(2 * d)).pre(2 * d);
  }
  return {begin(g), begin(g) + deg};
}

/*
 *  @brief 逆関数
 */





using namespace std;

template <typename Int, typename UInt, typename Long, typename ULong, int id>
struct ArbitraryLazyMontgomeryModIntBase {
  using mint = ArbitraryLazyMontgomeryModIntBase;

  inline static UInt mod;
  inline static UInt r;
  inline static UInt n2;
  static constexpr int bit_length = sizeof(UInt) * 8;

  static UInt get_r() {
    UInt ret = mod;
    while (mod * ret != 1) ret *= UInt(2) - mod * ret;
    return ret;
  }
  static void set_mod(UInt m) {
    assert(m < (UInt(1u) << (bit_length - 2)));
    assert((m & 1) == 1);
    mod = m, n2 = -ULong(m) % m, r = get_r();
  }
  UInt a;

  ArbitraryLazyMontgomeryModIntBase() : a(0) {}
  ArbitraryLazyMontgomeryModIntBase(const Long &b)
      : a(reduce(ULong(b % mod + mod) * n2)){};

  static UInt reduce(const ULong &b) {
    return (b + ULong(UInt(b) * UInt(-r)) * mod) >> bit_length;
  }

  mint &operator+=(const mint &b) {
    if (Int(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }
  mint &operator-=(const mint &b) {
    if (Int(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }
  mint &operator*=(const mint &b) {
    a = reduce(ULong(a) * b.a);
    return *this;
  }
  mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  mint operator+(const mint &b) const { return mint(*this) += b; }
  mint operator-(const mint &b) const { return mint(*this) -= b; }
  mint operator*(const mint &b) const { return mint(*this) *= b; }
  mint operator/(const mint &b) const { return mint(*this) /= b; }

  bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  mint operator-() const { return mint(0) - mint(*this); }
  mint operator+() const { return mint(*this); }

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

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

  friend istream &operator>>(istream &is, mint &b) {
    Long t;
    is >> t;
    b = ArbitraryLazyMontgomeryModIntBase(t);
    return (is);
  }

  mint inverse() const {
    Int x = get(), y = get_mod(), u = 1, v = 0;
    while (y > 0) {
      Int t = x / y;
      swap(x -= t * y, y);
      swap(u -= t * v, v);
    }
    return mint{u};
  }

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

  static UInt get_mod() { return mod; }
};

// id に適当な乱数を割り当てて使う
template <int id>
using ArbitraryLazyMontgomeryModInt =
    ArbitraryLazyMontgomeryModIntBase<int, unsigned int, long long,
                                      unsigned long long, id>;
template <int id>
using ArbitraryLazyMontgomeryModInt64bit =
    ArbitraryLazyMontgomeryModIntBase<long long, unsigned long long, __int128_t,
                                      __uint128_t, id>;


int64_t mod_sqrt(const int64_t &a, const int64_t &p) {
  assert(0 <= a && a < p);
  if (a < 2) return a;
  using Mint = ArbitraryLazyMontgomeryModInt<409075245>;
  Mint::set_mod(p);
  if (Mint(a).pow((p - 1) >> 1) != 1) return -1;
  Mint b = 1, one = 1;
  while (b.pow((p - 1) >> 1) == 1) b += one;
  int64_t m = p - 1, e = 0;
  while (m % 2 == 0) m >>= 1, e += 1;
  Mint x = Mint(a).pow((m - 1) >> 1);
  Mint y = Mint(a) * x * x;
  x *= a;
  Mint z = Mint(b).pow(m);
  while (y != 1) {
    int64_t j = 0;
    Mint t = y;
    while (t != one) {
      j += 1;
      t *= t;
    }
    z = z.pow(int64_t(1) << (e - j - 1));
    x *= z;
    z *= z;
    y *= z;
    e = j;
  }
  return x.get();
}

/**
 * @brief mod sqrt(Tonelli-Shanks algorithm)
 * @docs docs/modulo/mod-sqrt.md
 */


template <typename mint>
FormalPowerSeries<mint> sqrt(const FormalPowerSeries<mint> &f, int deg = -1) {
  if (deg == -1) deg = (int)f.size();
  if ((int)f.size() == 0) return FormalPowerSeries<mint>(deg, 0);
  if (f[0] == mint(0)) {
    for (int i = 1; i < (int)f.size(); i++) {
      if (f[i] != mint(0)) {
        if (i & 1) return {};
        if (deg - i / 2 <= 0) break;
        auto ret = sqrt(f >> i, deg - i / 2);
        if (ret.empty()) return {};
        ret = ret << (i / 2);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
    }
    return FormalPowerSeries<mint>(deg, 0);
  }

  int64_t sqr = mod_sqrt(f[0].get(), mint::get_mod());
  if (sqr == -1) return {};
  assert(sqr * sqr % mint::get_mod() == f[0].get());
  FormalPowerSeries<mint> ret = {mint(sqr)};
  mint inv2 = mint(2).inverse();
  for (int i = 1; i < deg; i <<= 1) {
    ret = (ret + f.pre(i << 1) * ret.inv(i << 1)) * inv2;
  }
  return ret.pre(deg);
}

/**
 * @brief 平方根
 * @docs docs/fps/fps-sqrt.md
 */


//

// #include "fps/ntt-friendly-fps.hpp"






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

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

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

  u32 a;

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

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

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

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

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

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

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

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

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

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

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

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

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


template <typename mint>
struct NTT {
  static constexpr uint32_t get_pr() {
    uint32_t _mod = mint::get_mod();
    using u64 = uint64_t;
    u64 ds[32] = {};
    int idx = 0;
    u64 m = _mod - 1;
    for (u64 i = 2; i * i <= m; ++i) {
      if (m % i == 0) {
        ds[idx++] = i;
        while (m % i == 0) m /= i;
      }
    }
    if (m != 1) ds[idx++] = m;

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

  static constexpr uint32_t mod = mint::get_mod();
  static constexpr uint32_t pr = get_pr();
  static constexpr int level = __builtin_ctzll(mod - 1);
  mint dw[level], dy[level];

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

  NTT() { setwy(level); }

  void fft4(vector<mint> &a, int k) {
    if ((int)a.size() <= 1) return;
    if (k == 1) {
      mint a1 = a[1];
      a[1] = a[0] - a[1];
      a[0] = a[0] + a1;
      return;
    }
    if (k & 1) {
      int v = 1 << (k - 1);
      for (int j = 0; j < v; ++j) {
        mint ajv = a[j + v];
        a[j + v] = a[j] - ajv;
        a[j] += ajv;
      }
    }
    int u = 1 << (2 + (k & 1));
    int v = 1 << (k - 2 - (k & 1));
    mint one = mint(1);
    mint imag = dw[1];
    while (v) {
      // jh = 0
      {
        int j0 = 0;
        int j1 = v;
        int j2 = j1 + v;
        int j3 = j2 + v;
        for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
          mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
          mint t0p2 = t0 + t2, t1p3 = t1 + t3;
          mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
          a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
          a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
        }
      }
      // jh >= 1
      mint ww = one, xx = one * dw[2], wx = one;
      for (int jh = 4; jh < u;) {
        ww = xx * xx, wx = ww * xx;
        int j0 = jh * v;
        int je = j0 + v;
        int j2 = je + v;
        for (; j0 < je; ++j0, ++j2) {
          mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww,
               t3 = a[j2 + v] * wx;
          mint t0p2 = t0 + t2, t1p3 = t1 + t3;
          mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
          a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
          a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
        }
        xx *= dw[__builtin_ctzll((jh += 4))];
      }
      u <<= 2;
      v >>= 2;
    }
  }

  void ifft4(vector<mint> &a, int k) {
    if ((int)a.size() <= 1) return;
    if (k == 1) {
      mint a1 = a[1];
      a[1] = a[0] - a[1];
      a[0] = a[0] + a1;
      return;
    }
    int u = 1 << (k - 2);
    int v = 1;
    mint one = mint(1);
    mint imag = dy[1];
    while (u) {
      // jh = 0
      {
        int j0 = 0;
        int j1 = v;
        int j2 = v + v;
        int j3 = j2 + v;
        for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
          mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
          mint t0p1 = t0 + t1, t2p3 = t2 + t3;
          mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
          a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
          a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
        }
      }
      // jh >= 1
      mint ww = one, xx = one * dy[2], yy = one;
      u <<= 2;
      for (int jh = 4; jh < u;) {
        ww = xx * xx, yy = xx * imag;
        int j0 = jh * v;
        int je = j0 + v;
        int j2 = je + v;
        for (; j0 < je; ++j0, ++j2) {
          mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
          mint t0p1 = t0 + t1, t2p3 = t2 + t3;
          mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
          a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
          a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
        }
        xx *= dy[__builtin_ctzll(jh += 4)];
      }
      u >>= 4;
      v <<= 2;
    }
    if (k & 1) {
      u = 1 << (k - 1);
      for (int j = 0; j < u; ++j) {
        mint ajv = a[j] - a[j + u];
        a[j] += a[j + u];
        a[j + u] = ajv;
      }
    }
  }

  void ntt(vector<mint> &a) {
    if ((int)a.size() <= 1) return;
    fft4(a, __builtin_ctz(a.size()));
  }

  void intt(vector<mint> &a) {
    if ((int)a.size() <= 1) return;
    ifft4(a, __builtin_ctz(a.size()));
    mint iv = mint(a.size()).inverse();
    for (auto &x : a) x *= iv;
  }

  vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
    int l = a.size() + b.size() - 1;
    if (min<int>(a.size(), b.size()) <= 40) {
      vector<mint> s(l);
      for (int i = 0; i < (int)a.size(); ++i)
        for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
      return s;
    }
    int k = 2, M = 4;
    while (M < l) M <<= 1, ++k;
    setwy(k);
    vector<mint> s(M);
    for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
    fft4(s, k);
    if (a.size() == b.size() && a == b) {
      for (int i = 0; i < M; ++i) s[i] *= s[i];
    } else {
      vector<mint> t(M);
      for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
      fft4(t, k);
      for (int i = 0; i < M; ++i) s[i] *= t[i];
    }
    ifft4(s, k);
    s.resize(l);
    mint invm = mint(M).inverse();
    for (int i = 0; i < l; ++i) s[i] *= invm;
    return s;
  }

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


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

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

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

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

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


template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
  ntt_ptr = nullptr;
}

template <typename mint>
void FormalPowerSeries<mint>::ntt() {
  exit(1);
}

template <typename mint>
void FormalPowerSeries<mint>::intt() {
  exit(1);
}

template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
  exit(1);
}

template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
  exit(1);
}

template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(
    const FormalPowerSeries<mint>& r) {
  if (this->empty() || r.empty()) {
    this->clear();
    return *this;
  }
  auto ret = ArbitraryNTT::multiply(*this, r);
  return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
  assert((*this)[0] != mint(0));
  if (deg == -1) deg = (*this).size();
  FormalPowerSeries<mint> ret({mint(1) / (*this)[0]});
  for (int i = 1; i < deg; i <<= 1)
    ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1);
  return ret.pre(deg);
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
  assert((*this).size() == 0 || (*this)[0] == mint(0));
  if (deg == -1) deg = (int)this->size();
  FormalPowerSeries<mint> ret({mint(1)});
  for (int i = 1; i < deg; i <<= 1) {
    ret = (ret * (pre(i << 1) + mint(1) - ret.log(i << 1))).pre(i << 1);
  }
  return ret.pre(deg);
}






//



using namespace std;



// g が sparse を仮定, f * g.inv() を計算
template <typename mint>
FormalPowerSeries<mint> sparse_div(const FormalPowerSeries<mint>& f,
                                   const FormalPowerSeries<mint>& g,
                                   int deg = -1) {
  assert(g.empty() == false && g[0] != mint(0));
  if (deg == -1) deg = f.size();
  mint ig0 = g[0].inverse();
  FormalPowerSeries<mint> s = f * ig0;
  s.resize(deg);
  vector<pair<int, mint>> gs;
  for (int i = 1; i < (int)g.size(); i++) {
    if (g[i] != 0) gs.emplace_back(i, g[i] * ig0);
  }
  for (int i = 0; i < deg; i++) {
    for (auto& [j, g_j] : gs) {
      if (i + j >= deg) break;
      s[i + j] -= s[i] * g_j;
    }
  }
  return s;
}

template <typename mint>
FormalPowerSeries<mint> sparse_inv(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() == false && f[0] != mint(0));
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }
  FormalPowerSeries<mint> g(deg);
  mint if0 = f[0].inverse();
  if (0 < deg) g[0] = if0;
  for (int k = 1; k < deg; k++) {
    for (auto& [j, fj] : fs) {
      if (k < j) break;
      g[k] += g[k - j] * fj;
    }
    g[k] *= -if0;
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_log(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() == false && f[0] == 1);
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  FormalPowerSeries<mint> g(deg);
  for (int k = 0; k < deg - 1; k++) {
    for (auto& [j, fj] : fs) {
      if (k < j) break;
      int i = k - j;
      g[k + 1] -= g[i + 1] * fj * (i + 1);
    }
    g[k + 1] *= invs[k + 1];
    if (k + 1 < (int)f.size()) g[k + 1] += f[k + 1];
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_exp(const FormalPowerSeries<mint>& f,
                                   int deg = -1) {
  assert(f.empty() or f[0] == 0);
  if (deg == -1) deg = f.size();
  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  FormalPowerSeries<mint> g(deg);
  if (deg) g[0] = 1;
  for (int k = 0; k < deg - 1; k++) {
    for (auto& [ip1, fip1] : fs) {
      int i = ip1 - 1;
      if (k < i) break;
      g[k + 1] += fip1 * g[k - i] * (i + 1);
    }
    g[k + 1] *= invs[k + 1];
  }
  return g;
}

template <typename mint>
FormalPowerSeries<mint> sparse_pow(const FormalPowerSeries<mint>& f,
                                   long long k, int deg = -1) {
  if (deg == -1) deg = f.size();
  if (k == 0) {
    FormalPowerSeries<mint> g(deg);
    if (deg) g[0] = 1;
    return g;
  }
  int zero = 0;
  while (zero != (int)f.size() and f[zero] == 0) zero++;
  if (zero == (int)f.size() or __int128_t(zero) * k >= deg) {
    return FormalPowerSeries<mint>(deg, 0);
  }
  if (zero != 0) {
    FormalPowerSeries<mint> suf{begin(f) + zero, end(f)};
    auto g = sparse_pow(suf, k, deg - zero * k);
    FormalPowerSeries<mint> h(zero * k, 0);
    copy(begin(g), end(g), back_inserter(h));
    return h;
  }

  int mod = mint::get_mod();
  static vector<mint> invs{1, 1};
  while ((int)invs.size() <= deg) {
    int i = invs.size();
    invs.push_back((-invs[mod % i]) * (mod / i));
  }

  vector<pair<int, mint>> fs;
  for (int i = 1; i < (int)f.size(); i++) {
    if (f[i] != 0) fs.emplace_back(i, f[i]);
  }

  FormalPowerSeries<mint> g(deg);
  g[0] = f[0].pow(k);
  mint denom = f[0].inverse();
  k %= mint::get_mod();
  for (int a = 1; a < deg; a++) {
    for (auto& [i, f_i] : fs) {
      if (a < i) break;
      g[a] += f_i * g[a - i] * ((k + 1) * i - a);
    }
    g[a] *= denom * invs[a];
  }
  return g;
}

/**
 * @brief sparse な形式的冪級数の演算
 */


//
using namespace Nyaan;
using mint = LazyMontgomeryModInt<1000000007>;
using vm = vector<mint>;
using vvm = vector<vm>;
Binomial<mint> C;
using fps = FormalPowerSeries<mint>;
using namespace Nyaan;

void test() {
  rep1(N, 10) {
    set<vvi> st, vis;
    auto add = [&](vvi v) {
      if (vis.count(v) == 0) {
        st.emplace(v);
        vis.emplace(v);
      }
    };
    vvi init;
    rep(i, N) init.push_back(vi{int(i)});
    add(init);

    vm cur(N + 1);
    while (sz(st)) {
      auto v = *begin(st);
      st.erase(v);
      cur[sz(v)] += 1;
      rep(i, sz(v) - 1) {
        {
          vvi w = v;
          vi a = w[i];
          vi b = w[i + 1];
          copy(all(b), back_inserter(a));
          w.erase(begin(w) + i + 1);
          w[i] = a;
          add(w);
        }
        {
          vvi w = v;
          vi a = w[i];
          vi b = w[i + 1];
          copy(all(a), back_inserter(b));
          w.erase(begin(w) + i + 1);
          w[i] = b;
          add(w);
        }
      }
    }
    trc2(N, cur);
  }
}

void q() {
  // test();

  inl(N, dummy, M);
  vl X(M);
  in(X);

#ifdef NyaanLocal
  {
    // https://oeis.org/A033877
    fps f = fps{1, -3} - sqrt(fps{1, -6, 1}, N + 6);
    f = f >> 1;
    f *= mint{2}.inverse();
    f += fps{1};
    f = f << 1;
    trc(f);
    rep1(i, N) { trc(f.pow(i)); }
    fps g = compositional_inverse(f);
    trc(g);
    trc(fps{0, 1, -1} * (fps{1, 1}.inv(N + 5)));
  }
#endif

  /*
  fps f(N + 1), g(N + 1);
  rep(i, N + 1) f[i] = C(N, i), g[i] = C(N, i) * (i % 2 ? -1 : 1);
  trc(f);
  trc(g);
  f *= g.inv();
  trc(f);
  */

  fps dp(N + 10);
  dp[0] = 1, dp[1] = N * 2;
  reg(n, 2, N + 1) {
    dp[n] = (dp[n - 1] * N * 2 + dp[n - 2] * (n - 2)) * C.inv(n);
  }
  trc(dp);
  each(K, X) { out(dp[N - K] * K * C.inv(N)); }
}

void Nyaan::solve() {
  int t = 1;
  // in(t);
  while (t--) q();
}