/**
 * date   : 2023-12-15 09:57:16
 * author : Nyaan
 */

#define NDEBUG

#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>


using namespace std;
namespace Nyaan {
template <typename T>
using V = vector<T>;
using vvi = V<V<int>>;
using ll = long long;
constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;

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

}  // namespace Nyaan



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

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



#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(); }

// #include "fps/polynomial-gcd.hpp"



using namespace std;

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

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

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


//


template <typename T>
struct edge {
  int src, to;
  T cost;

  edge(int _to, T _cost) : src(-1), to(_to), cost(_cost) {}
  edge(int _src, int _to, T _cost) : src(_src), to(_to), cost(_cost) {}

  edge &operator=(const int &x) {
    to = x;
    return *this;
  }

  operator int() const { return to; }
};
template <typename T>
using Edges = vector<edge<T>>;
template <typename T>
using WeightedGraph = vector<Edges<T>>;
using UnweightedGraph = vector<vector<int>>;

// Input of (Unweighted) Graph
UnweightedGraph graph(int N, int M = -1, bool is_directed = false,
                      bool is_1origin = true) {
  UnweightedGraph g(N);
  if (M == -1) M = N - 1;
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    if (is_1origin) x--, y--;
    g[x].push_back(y);
    if (!is_directed) g[y].push_back(x);
  }
  return g;
}

// Input of Weighted Graph
template <typename T>
WeightedGraph<T> wgraph(int N, int M = -1, bool is_directed = false,
                        bool is_1origin = true) {
  WeightedGraph<T> g(N);
  if (M == -1) M = N - 1;
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    T c;
    cin >> c;
    if (is_1origin) x--, y--;
    g[x].emplace_back(x, y, c);
    if (!is_directed) g[y].emplace_back(y, x, c);
  }
  return g;
}

// Input of Edges
template <typename T>
Edges<T> esgraph(int N, int M, int is_weighted = true, bool is_1origin = true) {
  Edges<T> es;
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    T c;
    if (is_weighted)
      cin >> c;
    else
      c = 1;
    if (is_1origin) x--, y--;
    es.emplace_back(x, y, c);
  }
  return es;
}

// Input of Adjacency Matrix
template <typename T>
vector<vector<T>> adjgraph(int N, int M, T INF, int is_weighted = true,
                           bool is_directed = false, bool is_1origin = true) {
  vector<vector<T>> d(N, vector<T>(N, INF));
  for (int _ = 0; _ < M; _++) {
    int x, y;
    cin >> x >> y;
    T c;
    if (is_weighted)
      cin >> c;
    else
      c = 1;
    if (is_1origin) x--, y--;
    d[x][y] = c;
    if (!is_directed) d[y][x] = c;
  }
  return d;
}

/**
 * @brief グラフテンプレート
 * @docs docs/graph/graph-template.md
 */






// 一般のグラフのstからの距離！！！！
// unvisited nodes : d = -1
vector<int> Depth(const UnweightedGraph &g, int start = 0) {
  int n = g.size();
  vector<int> ds(n, -1);
  ds[start] = 0;
  queue<int> q;
  q.push(start);
  while (!q.empty()) {
    int c = q.front();
    q.pop();
    int dc = ds[c];
    for (auto &d : g[c]) {
      if (ds[d] == -1) {
        ds[d] = dc + 1;
        q.push(d);
      }
    }
  }
  return ds;
}

// Depth of Rooted Weighted Tree
// unvisited nodes : d = -1
template <typename T>
vector<T> Depth(const WeightedGraph<T> &g, int start = 0) {
  vector<T> d(g.size(), -1);
  auto dfs = [&](auto rec, int cur, T val, int par = -1) -> void {
    d[cur] = val;
    for (auto &dst : g[cur]) {
      if (dst == par) continue;
      rec(rec, dst, val + dst.cost, cur);
    }
  };
  dfs(dfs, start, 0);
  return d;
}

// Diameter of Tree
// return value : { {u, v}, length }
pair<pair<int, int>, int> Diameter(const UnweightedGraph &g) {
  auto d = Depth(g, 0);
  int u = max_element(begin(d), end(d)) - begin(d);
  d = Depth(g, u);
  int v = max_element(begin(d), end(d)) - begin(d);
  return make_pair(make_pair(u, v), d[v]);
}

// Diameter of Weighted Tree
// return value : { {u, v}, length }
template <typename T>
pair<pair<int, int>, T> Diameter(const WeightedGraph<T> &g) {
  auto d = Depth(g, 0);
  int u = max_element(begin(d), end(d)) - begin(d);
  d = Depth(g, u);
  int v = max_element(begin(d), end(d)) - begin(d);
  return make_pair(make_pair(u, v), d[v]);
}

// nodes on the path u-v ( O(N) )
template <typename G>
vector<int> Path(G &g, int u, int v) {
  vector<int> ret;
  int end = 0;
  auto dfs = [&](auto rec, int cur, int par = -1) -> void {
    ret.push_back(cur);
    if (cur == v) {
      end = 1;
      return;
    }
    for (int dst : g[cur]) {
      if (dst == par) continue;
      rec(rec, dst, cur);
      if (end) return;
    }
    if (end) return;
    ret.pop_back();
  };
  dfs(dfs, u);
  return ret;
}

/**
 * @brief グラフユーティリティ
 * @docs docs/graph/graph-utility.md
 */






template <typename T>
struct has_cost {
 private:
  template <typename U>
  static auto confirm(U u) -> decltype(u.cost, std::true_type());
  static auto confirm(...) -> std::false_type;

 public:
  enum : bool { value = decltype(confirm(std::declval<T>()))::value };
};

template <typename T>
vector<vector<T>> inverse_tree(const vector<vector<T>>& g) {
  int N = (int)g.size();
  vector<vector<T>> rg(N);
  for (int i = 0; i < N; i++) {
    for (auto& e : g[i]) {
      if constexpr (is_same<T, int>::value) {
        rg[e].push_back(i);
      } else if constexpr (has_cost<T>::value) {
        rg[e].emplace_back(e.to, i, e.cost);
      } else {
        assert(0);
      }
    }
  }
  return rg;
}

template <typename T>
vector<vector<T>> rooted_tree(const vector<vector<T>>& g, int root = 0) {
  int N = (int)g.size();
  vector<vector<T>> rg(N);
  vector<char> v(N, false);
  v[root] = true;
  queue<int> que;
  que.emplace(root);
  while (!que.empty()) {
    auto p = que.front();
    que.pop();
    for (auto& e : g[p]) {
      if (v[e] == false) {
        v[e] = true;
        que.push(e);
        rg[p].push_back(e);
      }
    }
  }
  return rg;
}

/**
 * @brief 根付き木・逆辺からなる木への変換
 */






template <typename G>
struct HeavyLightDecomposition {
 private:
  void dfs_sz(int cur) {
    size[cur] = 1;
    for (auto& dst : g[cur]) {
      if (dst == par[cur]) {
        if (g[cur].size() >= 2 && int(dst) == int(g[cur][0]))
          swap(g[cur][0], g[cur][1]);
        else
          continue;
      }
      depth[dst] = depth[cur] + 1;
      par[dst] = cur;
      dfs_sz(dst);
      size[cur] += size[dst];
      if (size[dst] > size[g[cur][0]]) {
        swap(dst, g[cur][0]);
      }
    }
  }

  void dfs_hld(int cur) {
    down[cur] = id++;
    for (auto dst : g[cur]) {
      if (dst == par[cur]) continue;
      nxt[dst] = (int(dst) == int(g[cur][0]) ? nxt[cur] : int(dst));
      dfs_hld(dst);
    }
    up[cur] = id;
  }

  // [u, v)
  vector<pair<int, int>> ascend(int u, int v) const {
    vector<pair<int, int>> res;
    while (nxt[u] != nxt[v]) {
      res.emplace_back(down[u], down[nxt[u]]);
      u = par[nxt[u]];
    }
    if (u != v) res.emplace_back(down[u], down[v] + 1);
    return res;
  }

  // (u, v]
  vector<pair<int, int>> descend(int u, int v) const {
    if (u == v) return {};
    if (nxt[u] == nxt[v]) return {{down[u] + 1, down[v]}};
    auto res = descend(u, par[nxt[v]]);
    res.emplace_back(down[nxt[v]], down[v]);
    return res;
  }

 public:
  G& g;
  int id;
  vector<int> size, depth, down, up, nxt, par;
  HeavyLightDecomposition(G& _g, int root = 0)
      : g(_g),
        id(0),
        size(g.size(), 0),
        depth(g.size(), 0),
        down(g.size(), -1),
        up(g.size(), -1),
        nxt(g.size(), root),
        par(g.size(), root) {
    dfs_sz(root);
    dfs_hld(root);
  }

  void build(int root) {
    dfs_sz(root);
    dfs_hld(root);
  }

  pair<int, int> idx(int i) const { return make_pair(down[i], up[i]); }

  template <typename F>
  void path_query(int u, int v, bool vertex, const F& f) {
    int l = lca(u, v);
    for (auto&& [a, b] : ascend(u, l)) {
      int s = a + 1, t = b;
      s > t ? f(t, s) : f(s, t);
    }
    if (vertex) f(down[l], down[l] + 1);
    for (auto&& [a, b] : descend(l, v)) {
      int s = a, t = b + 1;
      s > t ? f(t, s) : f(s, t);
    }
  }

  template <typename F>
  void path_noncommutative_query(int u, int v, bool vertex, const F& f) {
    int l = lca(u, v);
    for (auto&& [a, b] : ascend(u, l)) f(a + 1, b);
    if (vertex) f(down[l], down[l] + 1);
    for (auto&& [a, b] : descend(l, v)) f(a, b + 1);
  }

  template <typename F>
  void subtree_query(int u, bool vertex, const F& f) {
    f(down[u] + int(!vertex), up[u]);
  }

  int lca(int a, int b) {
    while (nxt[a] != nxt[b]) {
      if (down[a] < down[b]) swap(a, b);
      a = par[nxt[a]];
    }
    return depth[a] < depth[b] ? a : b;
  }

  int dist(int a, int b) { return depth[a] + depth[b] - depth[lca(a, b)] * 2; }
};

/**
 * @brief Heavy Light Decomposition(重軽分解)
 * @docs docs/tree/heavy-light-decomposition.md
 */


//


template <typename fps>
struct fps_fraction {
  using frac = fps_fraction;
  using mint = typename fps::value_type;

  fps p, q;
  fps_fraction(const fps& numerator = fps{0}, const fps& denominator = fps{1})
      : p(numerator), q(denominator) {}

  friend frac operator+(const frac& l, const frac& r) {
    return frac{l.p * r.q + r.p * l.q, l.q * r.q};
  }
  friend frac operator-(const frac& l, const frac& r) {
    return frac{l.p * r.q - r.p * l.q, l.q * r.q};
  }
  friend frac operator*(const frac& l, const frac& r) {
    return frac{l.p * r.p, l.q * r.q};
  }
  friend frac operator/(const frac& l, const frac& r) {
    return frac{l.p * r.q, l.q * r.p};
  }

  frac& operator+=(const mint& r) {
    (*this).p += (*this).q * r;
    return *this;
  }
  frac& operator-=(const mint& r) {
    (*this).p -= (*this).q * r;
    return *this;
  }
  frac& operator*=(const mint& r) {
    (*this).p *= r;
    return *this;
  }

  frac operator+(const mint& r) { return frac{*this} += r; }
  frac operator-(const mint& r) { return frac{*this} -= r; }
  frac operator*(const mint& r) { return frac{*this} *= r; }
  frac operator/(const mint& r) { return frac{*this} /= r; }
  frac& operator+=(const frac& r) { return *this = (*this) + r; }
  frac& operator-=(const frac& r) { return *this = (*this) - r; }
  frac& operator*=(const frac& r) { return *this = (*this) * r; }
  frac operator-() const { return frac{-p, q}; }
  frac inverse() const { return frac{q, p}; };

  void shrink() { p.shrink(), q.shrink(); }
  friend bool operator==(const frac& l, const frac& r) {
    return l.p == r.p && l.q == r.q;
  }
  friend bool operator!=(const frac& l, const frac& r) {
    return l.p != r.p || l.q != r.q;
  }
  friend ostream& operator<<(ostream& os, const frac& r) {
    os << "[ ";
    for (auto& x : r.p) os << x << ", ";
    os << "], ";
    os << "[ ";
    for (auto& x : r.q) os << x << ", ";
    os << " ]";
    return os;
  }
};





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
 */


template <typename fps>
fps Pi(vector<fps> v) {
  if ((int)v.size() == 0) return fps{1};
  while ((int)v.size() >= 2) {
    vector<fps> nx;
    for (int i = 0; i + 1 < (int)v.size(); i += 2)
      nx.push_back(v[i] * v[i + 1]);
    if (v.size() % 2) nx.push_back(v.back());
    v = nx;
  }
  return v.back();
}






template <typename mint>
mint LinearRecurrence(long long k, FormalPowerSeries<mint> Q,
                      FormalPowerSeries<mint> P) {
  Q.shrink();
  mint ret = 0;
  if (P.size() >= Q.size()) {
    auto R = P / Q;
    P -= R * Q;
    P.shrink();
    if (k < (int)R.size()) ret += R[k];
  }
  if ((int)P.size() == 0) return ret;

  FormalPowerSeries<mint>::set_fft();
  if (FormalPowerSeries<mint>::ntt_ptr == nullptr) {
    P.resize((int)Q.size() - 1);
    while (k) {
      auto Q2 = Q;
      for (int i = 1; i < (int)Q2.size(); i += 2) Q2[i] = -Q2[i];
      auto S = P * Q2;
      auto T = Q * Q2;
      if (k & 1) {
        for (int i = 1; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      } else {
        for (int i = 0; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
        for (int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
      }
      k >>= 1;
    }
    return ret + P[0];
  } else {
    int N = 1;
    while (N < (int)Q.size()) N <<= 1;

    P.resize(2 * N);
    Q.resize(2 * N);
    P.ntt();
    Q.ntt();
    vector<mint> S(2 * N), T(2 * N);

    vector<int> btr(N);
    for (int i = 0, logn = __builtin_ctz(N); i < (1 << logn); i++) {
      btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (logn - 1));
    }
    mint dw = mint(FormalPowerSeries<mint>::ntt_pr())
                  .inverse()
                  .pow((mint::get_mod() - 1) / (2 * N));

    while (k) {
      mint inv2 = mint(2).inverse();

      // even degree of Q(x)Q(-x)
      T.resize(N);
      for (int i = 0; i < N; i++) T[i] = Q[(i << 1) | 0] * Q[(i << 1) | 1];

      S.resize(N);
      if (k & 1) {
        // odd degree of P(x)Q(-x)
        for (auto &i : btr) {
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] -
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) *
                 inv2;
          inv2 *= dw;
        }
      } else {
        // even degree of P(x)Q(-x)
        for (int i = 0; i < N; i++) {
          S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] +
                  P[(i << 1) | 1] * Q[(i << 1) | 0]) *
                 inv2;
        }
      }
      swap(P, S);
      swap(Q, T);
      k >>= 1;
      if (k < N) break;
      P.ntt_doubling();
      Q.ntt_doubling();
    }
    P.intt();
    Q.intt();
    return ret + (P * (Q.inv()))[k];
  }
}

template <typename mint>
mint kitamasa(long long N, FormalPowerSeries<mint> Q,
              FormalPowerSeries<mint> a) {
  assert(!Q.empty() && Q[0] != 0);
  if (N < (int)a.size()) return a[N];
  assert((int)a.size() >= int(Q.size()) - 1);
  auto P = a.pre((int)Q.size() - 1) * Q;
  P.resize(Q.size() - 1);
  return LinearRecurrence<mint>(N, Q, P);
}

/**
 * @brief 線形漸化式の高速計算
 * @docs docs/fps/kitamasa.md
 */




template <typename T, int H, int W>
struct Matrix {
  using Array = array<array<T, W>, H>;
  Array A;

  Matrix() : A() {
    for (int i = 0; i < H; i++)
      for (int j = 0; j < W; j++) (*this)[i][j] = T();
  }

  int height() const { return H; }

  int width() const { return W; }

  inline const array<T, W> &operator[](int k) const { return A[k]; }

  inline array<T, W> &operator[](int k) { return A[k]; }

  static Matrix I() {
    assert(H == W);
    Matrix mat;
    for (int i = 0; i < H; i++) mat[i][i] = 1;
    return (mat);
  }

  Matrix &operator+=(const Matrix &B) {
    for (int i = 0; i < H; i++)
      for (int j = 0; j < W; j++) A[i][j] += B[i][j];
    return (*this);
  }

  Matrix &operator-=(const Matrix &B) {
    for (int i = 0; i < H; i++)
      for (int j = 0; j < W; j++) A[i][j] -= B[i][j];
    return (*this);
  }

  Matrix &operator*=(const Matrix &B) {
    assert(H == W);
    Matrix C;
    for (int i = 0; i < H; i++)
      for (int k = 0; k < H; k++)
        for (int j = 0; j < H; j++) C[i][j] += A[i][k] * B[k][j];
    A.swap(C.A);
    return (*this);
  }

  Matrix &operator^=(long long k) {
    Matrix B = Matrix::I();
    while (k > 0) {
      if (k & 1) B *= *this;
      *this *= *this;
      k >>= 1LL;
    }
    A.swap(B.A);
    return (*this);
  }

  Matrix operator+(const Matrix &B) const { return (Matrix(*this) += B); }

  Matrix operator-(const Matrix &B) const { return (Matrix(*this) -= B); }

  Matrix operator*(const Matrix &B) const { return (Matrix(*this) *= B); }

  Matrix operator^(const long long k) const { return (Matrix(*this) ^= k); }

  bool operator==(const Matrix &B) const {
    for (int i = 0; i < H; i++)
      for (int j = 0; j < W; j++)
        if (A[i][j] != B[i][j]) return false;
    return true;
  }

  bool operator!=(const Matrix &B) const {
    for (int i = 0; i < H; i++)
      for (int j = 0; j < W; j++)
        if (A[i][j] != B[i][j]) return true;
    return false;
  }

  friend ostream &operator<<(ostream &os,const Matrix &p) {
    for (int i = 0; i < H; i++) {
      os << "[";
      for (int j = 0; j < W; j++) {
        os << p[i][j] << (j + 1 == W ? "]\n" : ",");
      }
    }
    return (os);
  }

  T determinant(int n = -1) {
    if (n == -1) n = H;
    Matrix B(*this);
    T ret = 1;
    for (int i = 0; i < n; i++) {
      int idx = -1;
      for (int j = i; j < n; j++) {
        if (B[j][i] != 0) {
          idx = j;
          break;
        }
      }
      if (idx == -1) return 0;
      if (i != idx) {
        ret *= T(-1);
        swap(B[i], B[idx]);
      }
      ret *= B[i][i];
      T inv = T(1) / B[i][i];
      for (int j = 0; j < n; j++) {
        B[i][j] *= inv;
      }
      for (int j = i + 1; j < n; j++) {
        T a = B[j][i];
        if (a == 0) continue;
        for (int k = i; k < n; k++) {
          B[j][k] -= B[i][k] * a;
        }
      }
    }
    return (ret);
  }
};

/**
 * @brief 行列ライブラリ(std::array版)
 */


//




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));
  }
};


template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
  if (!ntt_ptr) ntt_ptr = new NTT<mint>;
}

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

template <typename mint>
void FormalPowerSeries<mint>::ntt() {
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}

template <typename mint>
void FormalPowerSeries<mint>::intt() {
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}

template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
  set_fft();
  static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}

template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
  set_fft();
  return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
  assert((*this)[0] != mint(0));
  if (deg == -1) deg = (int)this->size();
  FormalPowerSeries<mint> res(deg);
  res[0] = {mint(1) / (*this)[0]};
  for (int d = 1; d < deg; d <<= 1) {
    FormalPowerSeries<mint> f(2 * d), g(2 * d);
    for (int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j];
    for (int j = 0; j < d; j++) g[j] = res[j];
    f.ntt();
    g.ntt();
    for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
    f.intt();
    for (int j = 0; j < d; j++) f[j] = 0;
    f.ntt();
    for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
    f.intt();
    for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
  }
  return res.pre(deg);
}

template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
  using fps = FormalPowerSeries<mint>;
  assert((*this).size() == 0 || (*this)[0] == mint(0));
  if (deg == -1) deg = this->size();

  fps inv;
  inv.reserve(deg + 1);
  inv.push_back(mint(0));
  inv.push_back(mint(1));

  auto inplace_integral = [&](fps& F) -> void {
    const int n = (int)F.size();
    auto mod = mint::get_mod();
    while ((int)inv.size() <= n) {
      int i = inv.size();
      inv.push_back((-inv[mod % i]) * (mod / i));
    }
    F.insert(begin(F), mint(0));
    for (int i = 1; i <= n; i++) F[i] *= inv[i];
  };

  auto inplace_diff = [](fps& F) -> void {
    if (F.empty()) return;
    F.erase(begin(F));
    mint coeff = 1, one = 1;
    for (int i = 0; i < (int)F.size(); i++) {
      F[i] *= coeff;
      coeff += one;
    }
  };

  fps b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
  for (int m = 2; m < deg; m *= 2) {
    auto y = b;
    y.resize(2 * m);
    y.ntt();
    z1 = z2;
    fps z(m);
    for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
    z.intt();
    fill(begin(z), begin(z) + m / 2, mint(0));
    z.ntt();
    for (int i = 0; i < m; ++i) z[i] *= -z1[i];
    z.intt();
    c.insert(end(c), begin(z) + m / 2, end(z));
    z2 = c;
    z2.resize(2 * m);
    z2.ntt();
    fps x(begin(*this), begin(*this) + min<int>(this->size(), m));
    x.resize(m);
    inplace_diff(x);
    x.push_back(mint(0));
    x.ntt();
    for (int i = 0; i < m; ++i) x[i] *= y[i];
    x.intt();
    x -= b.diff();
    x.resize(2 * m);
    for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
    x.ntt();
    for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
    x.intt();
    x.pop_back();
    inplace_integral(x);
    for (int i = m; i < min<int>(this->size(), 2 * m); ++i) x[i] += (*this)[i];
    fill(begin(x), begin(x) + m, mint(0));
    x.ntt();
    for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
    x.intt();
    b.insert(end(b), begin(x) + m, end(x));
  }
  return fps{begin(b), begin(b) + deg};
}

/**
 * @brief NTT mod用FPSライブラリ
 * @docs docs/fps/ntt-friendly-fps.md
 */




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; }
};




// #include "fps/arbitrary-fps.hpp"
//
using namespace Nyaan;
using mint = LazyMontgomeryModInt<998244353>;
// using mint = LazyMontgomeryModInt<1000000007>;
using vm = vector<mint>;
using vvm = vector<vm>;
Binomial<mint> C;
using fps = FormalPowerSeries<mint>;
using namespace Nyaan;

using frac = fps_fraction<fps>;
using Mat = Matrix<fps, 2, 2>;

void shrink(frac& f) { f.p.shrink(), f.q.shrink(); }
void shrink(Mat& m) { rep(i, 2) rep(j, 2) m[i][j].shrink(); }

vvi g2;

frac light(int c);

V<frac> heavy(int c) {
  if (g2[c].empty()) return V<frac>{frac{fps{0}, fps{1}}};

  vector<frac> v, res = heavy(g2[c][0]);
  v.push_back(frac{fps{0}, fps{1}});
  rep1(i, sz(g2[c]) - 1) v.push_back(light(g2[c][i]));
  while (v.size() >= 2u) {
    vector<frac> w;
    for (int i = 0; i + 1 < (int)v.size(); i += 2) w.push_back(v[i] + v[i + 1]);
    if (v.size() & 1) w.push_back(v.back());
    each(x, w) shrink(x);
    swap(v, w);
  }
  res.push_back(v.back());
  /*
  vector<frac> res = heavy(g2[c][0]);

  auto frac_comp = [&](const frac& a, const frac& b) {
    return a.q.size() > b.q.size();
  };
  priority_queue<frac, vector<frac>, decltype(frac_comp)> Q(frac_comp);

  Q.push(frac{fps{0}, fps{1}});
  rep1(i, sz(g2[c]) - 1) Q.push(light(g2[c][i]));
  while (Q.size() >= 2u) {
    auto f0 = Q.top();
    Q.pop();
    auto f1 = Q.top();
    Q.pop();
    Q.push(f0 + f1);
  }
  res.push_back(Q.top());
  */
  return res;
}

frac light(int c) {
  auto v = heavy(c);

  Mat I;
  I[0][0] = I[1][1] = fps{1};
  /*
  Mat oya(2);
  oya[0][1] = fps{1};
  oya[1][0] = fps{0, 0, -1};
  oya[1][1] = fps{1, -1};
  */

  V<Mat> ms{I};
  each(f, v) {
    Mat m;
    /*
    m[0][0] = m[1][1] = f.q;
    m[0][1] = f.p;
    ms.push_back(m);
    ms.push_back(oya);
    */
    shrink(f);
    m[0][1] = f.q;
    m[1][0] = -f.q << 2;
    m[1][1] = (-f.p << 2) + f.q - (f.q << 1);
    ms.push_back(m);
  }
  reverse(all(ms));
  while (sz(ms) >= 2) {
    V<Mat> nx;
    for (int i = 0; i + 1 < sz(ms); i += 2) {
      nx.push_back(ms[i] * ms[i + 1]);
    }
    if (sz(ms) % 2) nx.push_back(ms.back());
    each(x, nx) shrink(x);
    ms = nx;
  }
  auto m = ms[0];
  fps p = m[0][1];
  fps q = m[1][1];
  return frac{p, q};
}

mint calc(int N, int M, int S, int T, vvi g) {
  auto path = Path(g, S, T);
  g = rooted_tree(g, T);
  HeavyLightDecomposition hld{g, T};
  if (sz(path) - 1 > M) return 0;

  {
    set<int> pathset;
    each(p, path) pathset.insert(p);
    g2.resize(N);
    rep(i, N) each(j, g[i]) {
      if (pathset.count(j) == 0) g2[i].push_back(j);
    }
  }

  // S, ..., T について問題を解く
  V<frac> dp(sz(path));
  rep(j, sz(path)) {
    int c = path[j];
    V<frac> v;
    v.push_back(frac{fps{0}, fps{1}});
    rep(i, sz(g2[c])) v.push_back(light(g2[c][i]));
    while (v.size() >= 2u) {
      vector<frac> w;
      for (int i = 0; i + 1 < (int)v.size(); i += 2)
        w.push_back(v[i] + v[i + 1]);
      if (v.size() & 1) w.push_back(v.back());
      each(x, w) shrink(x);
      swap(v, w);
    }
    dp[j] = v[0];
    shrink(dp[j]);
  }

  /*
  // 一旦愚直を書く
  frac prod{fps{1}, fps{1}};
  {
    frac s{fps{0}, fps{1}};
    each(x, dp) {
      s += x;
      s = fps{1, -1} - s * fps{0, 0, 1};
      swap(s.p, s.q);
      prod *= s;
    }
  }
  return LinearRecurrence(M - (sz(path) - 1), prod.q, prod.p);
  */

  trc2("dp ok");
  {
    ll dsum = 0;
    each(x, dp) dsum += sz(x.p) + sz(x.q);
    trc2(dsum);
  }

  frac fs = fps{1, -1} - dp[0] * fps{0, 0, 1};
  swap(fs.p, fs.q);

  V<Mat> ms;
  {
    Mat m;
    m[0][0] = m[1][1] = fps{1};
    ms.push_back(m);
  }
  rep1(i, sz(path) - 1) {
    Mat m;
    m[0][1] = dp[i].q;
    m[1][0] = dp[i].q * fps{0, 0, -1};
    m[1][1] = dp[i].q * fps{1, -1} - dp[i].p * fps{0, 0, 1};
    ms.push_back(m);
  }
  reverse(all(ms));
  while (sz(ms) >= 2) {
    V<Mat> nx;
    for (int i = 0; i + 1 < sz(ms); i += 2) nx.push_back(ms[i] * ms[i + 1]);
    if (sz(ms) % 2) nx.push_back(ms.back());
    each(x, nx) shrink(x);
    ms = nx;
  }

  auto m = ms[0];
  V<fps> ftps;
  ftps.push_back(fs.p);
  rep1(i, sz(path) - 1) ftps.push_back(dp[i].q);
  fps ftp = Pi(ftps);
  fps ftq = fs.p * m[1][0] + fs.q * m[1][1];
  trc2("ft ok");
  trc2(sz(ftp), sz(ftq));
  return LinearRecurrence(M - (sz(path) - 1), ftq, ftp);
}

void q() {
  Timer timer;
  int N, M, S, T;
  cin >> N >> M >> S >> T;
  auto g = graph(N);
  S--, T--;
  // out(naive(N, M, S, T, g));
  cout << calc(N, M, S, T, g) << "\n";
  trc2(timer());
}

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