#pragma region kyopro_template
#include <bits/stdc++.h>
#define pb push_back
#define eb emplace_back
#define fi first
#define se second
#define each(x, v) for (auto &x : v)
#define all(v) (v).begin(), (v).end()
#define sz(v) ((int)(v).size())
#define mem(a, val) memset(a, val, sizeof(a))
#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 inc(...)    \
  char __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 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 die(...)      \
  do {                \
    out(__VA_ARGS__); \
    return;           \
  } while (0)

using namespace std;
using ll = long long;
template <class T>
using V = vector<T>;
using vi = vector<int>;
using vl = vector<long long>;
using vvi = vector<vector<int>>;
using vd = V<double>;
using vs = V<string>;
using vvl = vector<vector<long long>>;
using P = pair<long long, long long>;
using vp = vector<P>;
using pii = pair<int, int>;
using vpi = vector<pair<int, int>>;
constexpr int inf = 1001001001;
constexpr long long infLL = (1LL << 61) - 1;
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, typename U>
T ceil(T a, U b) {
  return (a + b - 1) / b;
}
constexpr long long TEN(int n) {
  long long ret = 1, x = 10;
  while (n) {
    if (n & 1) ret *= x;
    x *= x;
    n >>= 1;
  }
  return ret;
}
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;
}
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>
void out(const T &t, const U &... u) {
  cout << t;
  if (sizeof...(u)) cout << " ";
  out(u...);
}

void solve();
#ifdef NyaanDebug
#include "NyaanDebug.h"
#define trc(...)                   \
  do {                             \
    cerr << #__VA_ARGS__ << " = "; \
    dbg_out(__VA_ARGS__);          \
  } while (0)
#define trca(v, N)       \
  do {                   \
    cerr << #v << " = "; \
    array_out(v, N);     \
  } while (0)
#define trcc(v)                             \
  do {                                      \
    cerr << #v << " = {";                   \
    each(x, v) { cerr << " " << x << ","; } \
    cerr << "}" << endl;                    \
  } while (0)
#else
#define trc(...)
#define trca(...)
#define trcc(...)
int main() { solve(); }
#endif

inline int popcnt(unsigned long long a) { return __builtin_popcountll(a); }
inline int lsb(unsigned long long a) { return __builtin_ctzll(a); }
inline int msb(unsigned long long a) { return 63 - __builtin_clzll(a); }
template <typename T>
inline int getbit(T a, int i) {
  return (a >> i) & 1;
}
template <typename T>
inline void setbit(T &a, int i) {
  a |= (1LL << i);
}
template <typename T>
inline void delbit(T &a, int i) {
  a &= ~(1LL << i);
}
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);
}
template <typename T>
vector<T> mkrui(const vector<T> &v) {
  vector<T> ret(v.size() + 1);
  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 = int>
vector<T> mkiota(int N) {
  vector<T> ret(N);
  iota(begin(ret), end(ret), 0);
  return ret;
}
template <typename T = int>
vector<T> mkinv(vector<T> &v) {
  vector<T> inv(v.size());
  for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
  return inv;
}

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

#pragma endregion

constexpr int MOD = /**/ 998244353;  //*/ 1000000007;

template <int mod>
struct ModInt {
  int x;

  ModInt() : x(0) {}

  ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

  ModInt &operator+=(const ModInt &p) {
    if ((x += p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator-=(const ModInt &p) {
    if ((x += mod - p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator*=(const ModInt &p) {
    x = (int)(1LL * x * p.x % mod);
    return *this;
  }

  ModInt &operator/=(const ModInt &p) {
    *this *= p.inverse();
    return *this;
  }

  ModInt operator-() const { return ModInt(-x); }

  ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; }

  ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; }

  ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; }

  ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; }

  bool operator==(const ModInt &p) const { return x == p.x; }

  bool operator!=(const ModInt &p) const { return x != p.x; }

  ModInt inverse() const {
    int a = x, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ModInt(u);
  }

  ModInt pow(int64_t n) const {
    ModInt ret(1), mul(x);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  friend ostream &operator<<(ostream &os, const ModInt &p) { return os << p.x; }

  friend istream &operator>>(istream &is, ModInt &a) {
    int64_t t;
    is >> t;
    a = ModInt<mod>(t);
    return (is);
  }

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

using modint = ModInt<MOD>;
using mint = modint;
using vm = vector<mint>;

namespace FastFourierTransform {
using real = double;

struct C {
  real x, y;

  C() : x(0), y(0) {}

  C(real x, real y) : x(x), y(y) {}

  inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }

  inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }

  inline C operator*(const C &c) const {
    return C(x * c.x - y * c.y, x * c.y + y * c.x);
  }

  inline C conj() const { return C(x, -y); }
};

const real PI = acosl(-1);
int base = 1;
vector<C> rts = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};

void ensure_base(int nbase) {
  if (nbase <= base) return;
  rev.resize(1 << nbase);
  rts.resize(1 << nbase);
  for (int i = 0; i < (1 << nbase); i++) {
    rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
  }
  while (base < nbase) {
    real angle = PI * 2.0 / (1 << (base + 1));
    for (int i = 1 << (base - 1); i < (1 << base); i++) {
      rts[i << 1] = rts[i];
      real angle_i = angle * (2 * i + 1 - (1 << base));
      rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
    }
    ++base;
  }
}

void fft(vector<C> &a, int n) {
  assert((n & (n - 1)) == 0);
  int zeros = __builtin_ctz(n);
  ensure_base(zeros);
  int shift = base - zeros;
  for (int i = 0; i < n; i++) {
    if (i < (rev[i] >> shift)) {
      swap(a[i], a[rev[i] >> shift]);
    }
  }
  for (int k = 1; k < n; k <<= 1) {
    for (int i = 0; i < n; i += 2 * k) {
      for (int j = 0; j < k; j++) {
        C z = a[i + j + k] * rts[j + k];
        a[i + j + k] = a[i + j] - z;
        a[i + j] = a[i + j] + z;
      }
    }
  }
}

vector<int64_t> multiply(const vector<int> &a, const vector<int> &b) {
  int need = (int)a.size() + (int)b.size() - 1;
  int nbase = 1;
  while ((1 << nbase) < need) nbase++;
  ensure_base(nbase);
  int sz = 1 << nbase;
  vector<C> fa(sz);
  for (int i = 0; i < sz; i++) {
    int x = (i < (int)a.size() ? a[i] : 0);
    int y = (i < (int)b.size() ? b[i] : 0);
    fa[i] = C(x, y);
  }
  fft(fa, sz);
  C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
  for (int i = 0; i <= (sz >> 1); i++) {
    int j = (sz - i) & (sz - 1);
    C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
    fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
    fa[i] = z;
  }
  for (int i = 0; i < (sz >> 1); i++) {
    C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
    C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
    fa[i] = A0 + A1 * s;
  }
  fft(fa, sz >> 1);
  vector<int64_t> ret(need);
  for (int i = 0; i < need; i++) {
    ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
  }
  return ret;
}
};  // namespace FastFourierTransform

template <typename T>
struct ArbitraryModConvolution {
  using real = FastFourierTransform::real;
  using C = FastFourierTransform::C;

  ArbitraryModConvolution() = default;

  vector<T> multiply(const vector<T> &a, const vector<T> &b, int need = -1) {
    if (need == -1) need = a.size() + b.size() - 1;
    int nbase = 0;
    while ((1 << nbase) < need) nbase++;
    FastFourierTransform::ensure_base(nbase);
    int sz = 1 << nbase;
    vector<C> fa(sz);
    for (int i = 0; i < (int)a.size(); i++) {
      fa[i] = C(a[i].x & ((1 << 15) - 1), a[i].x >> 15);
    }
    fft(fa, sz);
    vector<C> fb(sz);
    if (a == b) {
      fb = fa;
    } else {
      for (int i = 0; i < (int)b.size(); i++) {
        fb[i] = C(b[i].x & ((1 << 15) - 1), b[i].x >> 15);
      }
      fft(fb, sz);
    }
    real ratio = 0.25 / sz;
    C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1);
    for (int i = 0; i <= (sz >> 1); i++) {
      int j = (sz - i) & (sz - 1);
      C a1 = (fa[i] + fa[j].conj());
      C a2 = (fa[i] - fa[j].conj()) * r2;
      C b1 = (fb[i] + fb[j].conj()) * r3;
      C b2 = (fb[i] - fb[j].conj()) * r4;
      if (i != j) {
        C c1 = (fa[j] + fa[i].conj());
        C c2 = (fa[j] - fa[i].conj()) * r2;
        C d1 = (fb[j] + fb[i].conj()) * r3;
        C d2 = (fb[j] - fb[i].conj()) * r4;
        fa[i] = c1 * d1 + c2 * d2 * r5;
        fb[i] = c1 * d2 + c2 * d1;
      }
      fa[j] = a1 * b1 + a2 * b2 * r5;
      fb[j] = a1 * b2 + a2 * b1;
    }
    fft(fa, sz);
    fft(fb, sz);
    vector<T> ret(need);
    for (int i = 0; i < need; i++) {
      int64_t aa = llround(fa[i].x);
      int64_t bb = llround(fb[i].x);
      int64_t cc = llround(fa[i].y);
      aa = T(aa).x, bb = T(bb).x, cc = T(cc).x;
      ret[i] = aa + (bb << 15) + (cc << 30);
    }
    return ret;
  }
};

template <int mod>
struct NumberTheoreticTransform {
  int base, max_base, root;
  vector<int> rev, rts;

  NumberTheoreticTransform() : base(1), rev{0, 1}, rts{0, 1} {
    assert(mod >= 3 && mod % 2 == 1);
    auto tmp = mod - 1;
    max_base = 0;
    while (tmp % 2 == 0) tmp >>= 1, max_base++;
    root = 2;
    while (mod_pow(root, (mod - 1) >> 1) == 1) ++root;
    assert(mod_pow(root, mod - 1) == 1);
    root = mod_pow(root, (mod - 1) >> max_base);
  }

  inline int mod_pow(int x, int n) {
    int ret = 1;
    while (n > 0) {
      if (n & 1) ret = mul(ret, x);
      x = mul(x, x);
      n >>= 1;
    }
    return ret;
  }

  inline int inverse(int x) { return mod_pow(x, mod - 2); }

  inline unsigned add(unsigned x, unsigned y) {
    x += y;
    if (x >= mod) x -= mod;
    return x;
  }

  inline unsigned mul(unsigned a, unsigned b) {
    return 1ull * a * b % (unsigned long long)mod;
  }

  void ensure_base(int nbase) {
    if (nbase <= base) return;
    rev.resize(1 << nbase);
    rts.resize(1 << nbase);
    for (int i = 0; i < (1 << nbase); i++) {
      rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
    }
    assert(nbase <= max_base);
    while (base < nbase) {
      int z = mod_pow(root, 1 << (max_base - 1 - base));
      for (int i = 1 << (base - 1); i < (1 << base); i++) {
        rts[i << 1] = rts[i];
        rts[(i << 1) + 1] = mul(rts[i], z);
      }
      ++base;
    }
  }

  void ntt(vector<int> &a) {
    const int n = (int)a.size();
    assert((n & (n - 1)) == 0);
    int zeros = __builtin_ctz(n);
    ensure_base(zeros);
    int shift = base - zeros;
    for (int i = 0; i < n; i++) {
      if (i < (rev[i] >> shift)) {
        swap(a[i], a[rev[i] >> shift]);
      }
    }
    for (int k = 1; k < n; k <<= 1) {
      for (int i = 0; i < n; i += 2 * k) {
        for (int j = 0; j < k; j++) {
          int z = mul(a[i + j + k], rts[j + k]);
          a[i + j + k] = add(a[i + j], mod - z);
          a[i + j] = add(a[i + j], z);
        }
      }
    }
  }

  vector<int> multiply(vector<int> a, vector<int> b) {
    int need = a.size() + b.size() - 1;
    int nbase = 1;
    while ((1 << nbase) < need) nbase++;
    ensure_base(nbase);
    int sz = 1 << nbase;
    a.resize(sz, 0);
    b.resize(sz, 0);
    ntt(a);
    ntt(b);
    int inv_sz = inverse(sz);
    for (int i = 0; i < sz; i++) {
      a[i] = mul(a[i], mul(b[i], inv_sz));
    }
    reverse(a.begin() + 1, a.end());
    ntt(a);
    a.resize(need);
    return a;
  }

  vector<modint> multiply_for_fps(const vector<modint> &a,
                                  const vector<modint> &b) {
    vector<int> A(a.size()), B(b.size());
    for (int i = 0; i < (int)a.size(); i++) A[i] = a[i].x;
    for (int i = 0; i < (int)b.size(); i++) B[i] = b[i].x;
    auto C = multiply(A, B);
    vector<modint> ret(C.size());
    for (int i = 0; i < (int)C.size(); i++) ret[i].x = C[i];
    return ret;
  }
};

template <typename T>
struct FormalPowerSeries : vector<T> {
  using vector<T>::vector;
  using P = FormalPowerSeries;

  using MULT = function<P(P, P)>;

  static MULT &get_mult() {
    static MULT mult = nullptr;
    return mult;
  }

  static void set_fft(MULT f) { get_mult() = f; }

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

  P operator+(const P &r) const { return P(*this) += r; }

  P operator+(const T &v) const { return P(*this) += v; }

  P operator-(const P &r) const { return P(*this) -= r; }

  P operator-(const T &v) const { return P(*this) -= v; }

  P operator*(const P &r) const { return P(*this) *= r; }

  P operator*(const T &v) const { return P(*this) *= v; }

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

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

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

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

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

  P &operator*=(const P &r) {
    if (this->empty() || r.empty()) {
      this->clear();
      return *this;
    }
    assert(get_mult() != nullptr);
    return *this = get_mult()(*this, r);
  }

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

  P operator-() const {
    P ret(this->size());
    for (int i = 0; i < this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }

  P &operator/=(const P &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
  }

  P pre(int sz) const {
    return P(begin(*this), begin(*this) + min((int)this->size(), sz));
  }

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

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

  P rev(int deg = -1) const {
    P ret(*this);
    if (deg != -1) ret.resize(deg, T(0));
    reverse(begin(ret), end(ret));
    return ret;
  }

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

  P integral() const {
    const int n = (int)this->size();
    P ret(n + 1);
    ret[0] = T(0);
    for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
    return ret;
  }

  // F(0) must not be 0
  P inv(int deg = -1) const {
    assert(((*this)[0]) != T(0));
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    P ret({T(1) / (*this)[0]});
    for (int i = 1; i < deg; i <<= 1) {
      ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1);
    }
    return ret.pre(deg);
  }

  // F(0) must be 1
  P log(int deg = -1) const {
    assert((*this)[0] == 1);
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }

  P sqrt(int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;

    if ((*this)[0] == T(0)) {
      for (int i = 1; i < n; i++) {
        if ((*this)[i] != T(0)) {
          if (i & 1) return {};
          if (deg - i / 2 <= 0) break;
          auto ret = (*this >> i).sqrt(deg - i / 2) << (i / 2);
          if (ret.size() < deg) ret.resize(deg, T(0));
          return ret;
        }
      }
      return P(deg, 0);
    }

    P ret({T(1)});
    T inv2 = T(1) / T(2);
    for (int i = 1; i < deg; i <<= 1) {
      ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
    }
    return ret.pre(deg);
  }

  // F(0) must be 0
  P exp(int deg = -1) const {
    assert((*this)[0] == T(0));
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    P ret({T(1)});
    for (int i = 1; i < deg; i <<= 1) {
      ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1);
    }
    return ret.pre(deg);
  }

  P pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    for (int i = 0; i < n; i++) {
      if ((*this)[i] != T(0)) {
        T rev = T(1) / (*this)[i];
        P C(*this * rev);
        P D(n - i);
        for (int j = i; j < n; j++) D[j - i] = C[j];
        D = (D.log() * k).exp() * (*this)[i].pow(k);
        P E(deg);
        if (i * k > deg) return E;
        auto S = i * k;
        for (int j = 0; j + S < deg && j < D.size(); j++) E[j + S] = D[j];
        return E;
      }
    }
    return *this;
  }

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

using FPS = FormalPowerSeries<modint>;

// fにa * x^n + bを掛ける
void mul_simple(FPS &f, modint a, int n, modint b) {
  for (int i = (int)f.size() - 1; i >= 0; i--) {
    f[i] *= b;
    if (i >= n) f[i] += f[i - n] * a;
  }
}

// fからa * x^n + bを割る
void div_simple(FPS &f, modint a, int n, modint b) {
  for (int i = 0; i < (int)f.size(); i++) {
    f[i] /= b;
    if (i + n < (int)f.size()) f[n + i] -= f[i] * a;
  }
}

// f / gをdeg(f)次まで求める
FPS div_(FPS &f, FPS g) {
  int n = f.size();
  return (f * g.inv(n)).pre(n);
}

modint kitamasa(ll N, FPS &Q, FPS &a) {
  int k = Q.size() - 1;
  assert((int)a.size() == k);
  FPS P = a * Q;
  P.resize(k);
  while (N) {
    auto Q2 = Q;
    for (int i = 1; i < (int)Q2.size(); i += 2) Q2[i].x = MOD - Q2[i].x;
    auto S = P * Q2;
    auto T = Q * Q2;
    if (N & 1) {
      for (int i = 1; i < (int)S.size(); i += 2) P[i >> 1].x = S[i].x;
      for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1].x = T[i].x;
    } else {
      for (int i = 0; i < (int)S.size(); i += 2) P[i >> 1].x = S[i].x;
      for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1].x = T[i].x;
    }
    N >>= 1;
  }
  return P[0];
}

struct DynamicUnionFind {
  unordered_map<int, int> m;
  DynamicUnionFind() {}

  int data(int k) {
    auto itk = m.find(k);
    return itk == m.end() ? m[k] = -1 : itk->second;
  }
  int find(int k) {
    int datk = data(k);
    return datk < 0 ? k : m[k] = find(datk);
  }

  int unite(int x, int y) {
    if ((x = find(x)) == (y = find(y))) return false;
    auto itx = m.find(x), ity = m.find(y);
    if (itx->second > ity->second) {
      ity->second += itx->second;
      itx->second = y;
    } else {
      itx->second += ity->second;
      ity->second = x;
    }
    return true;
  }

  int size(int k) { return -data(find(k)); }

  int same(int x, int y) { return find(x) == find(y); }
};

vector<ll> fac, finv, inv;
void cominit(int MAX) {
  MAX++;
  fac.resize(MAX, 0);
  finv.resize(MAX, 0);
  inv.resize(MAX, 0);
  fac[0] = fac[1] = finv[0] = finv[1] = inv[1] = 1;
  for (int i = 2; i < MAX; i++) {
    fac[i] = fac[i - 1] * i % MOD;
    inv[i] = MOD - inv[MOD % i] * (MOD / i) % MOD;
    finv[i] = finv[i - 1] * inv[i] % MOD;
  }
}
// nCk combination
inline long long COM(int n, int k) {
  if (n < k || k < 0 || n < 0)
    return 0;
  else
    return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD;
}
// nPk permutation
inline long long PER(int n, int k) {
  if (n < k || k < 0 || n < 0)
    return 0;
  else
    return (fac[n] * finv[n - k]) % MOD;
}
// nHk homogeneous polynomial
inline long long HGP(int n, int k) {
  if (n == 0 && k == 0)
    return 1;  // depending on problem?
  else if (n < 1 || k < 0)
    return 0;
  else
    return fac[n + k - 1] * (finv[k] * finv[n - 1] % MOD) % MOD;
}

void solve() {
  NumberTheoreticTransform<MOD> ntt;
  auto mul = [&](const FPS::P &a, const FPS::P &b) {
    auto ret = ntt.multiply_for_fps(a, b);
    return FPS::P(ret.begin(), ret.end());
  };
  FPS::set_fft(mul);

  ini(N, M);
  vp e(M);
  in(e);
  each(p, e) p.first--, p.second--;

  DynamicUnionFind uf;
  unordered_map<int, int> m;

  cominit(N * 2 + 100);

  map<P, mint> memo;

  mint ans = 0;
  // houzyo
  rep(b, 1 << M) {
    uf.m.clear();
    m.clear();

    vi ps;
    rep(i, M) {
      if (!getbit(b, i)) continue;
      m[e[i].first]++, m[e[i].second]++;
      uf.unite(e[i].first, e[i].second);
      ps.pb(e[i].first);
      ps.pb(e[i].second);
    }

    int flg = 0;
    each(x, m) if (x.second >= 3) flg = 1;
    if (flg) continue;

    ps = mkuni(ps);
    vi used(sz(ps));
    // 連結成分ごとにアをする めんどいな

    int cycle = 0, notcycle = 0, dec = 0;
    rep(i, sz(ps)) {
      if (used[i]) continue;
      // 同じ連結成分は？
      vi p{ps[i]};
      used[i] = 1;
      for (int j = i + 1; j < sz(ps); j++)
        if (uf.same(ps[i], ps[j])) p.pb(ps[j]), used[j] = 1;
      // saikuru?
      assert(p.size()==uf.size(p.back()));
      int all2 = 1;
      each(x, p) if (m[x] != 2) all2 = 0;
      if (all2)
        cycle += 1;
      else
        notcycle += 1, dec += sz(p);
    }

    mint coeff = popcnt(b) % 2 == 0 ? 1 : -1;
    if (cycle >= 2 or (cycle == 1 and notcycle)) continue;
    if (cycle == 1) {
      ans += coeff;
      continue;
    }
    // soreigai
    // 絶対使う向きつき頂点 notcycle個
    // 使わなくてもいい点 N-dec個

    trc(cycle, notcycle, dec);
    // 1ko mukikotei
    mint cur = 1;
    if (notcycle >= 2) cur *= mint(2).pow(notcycle - 1) * fac[notcycle - 1];

    // soreigai ireru
    int M = N - dec;

    //
    // 1ko
    // hazimeha notcycle
    // i ko tukau
    // no * (no+1) * ... * (no + i - 1) * NCi
    // koreha? 1st starling?

    if (memo.find(P{notcycle, dec}) != memo.end()) {
      cur *= memo[P{notcycle, dec}];
    } else {
      mint x = 0;
      if (notcycle == 0) {
        // N choten
        for (int i = 3; i <= N; i++) {
          x += mint(COM(N, i)) * mint(fac[i - 1]) / 2;
        }

      } else {
        for (int i = 0; i <= M; i++) {
          if (dec == 2 and i == 0)
            continue;
          else
            x += mint(COM(M, i)) * fac[notcycle + i - 1] * finv[notcycle - 1];
        }
      }
      memo[P{notcycle, dec}] = x;
      cur *= x;
    }
    ans += cur * coeff;
  }
  out(ans);
}