#line 1 "/home/maspy/compro/library/my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else

// https://codeforces.com/blog/entry/96344
#pragma GCC optimize("Ofast,unroll-loops")
// いまの CF だとこれ入れると動かない？
// #pragma GCC target("avx2,popcnt")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'010'000'000;
template <>
constexpr ll infty<ll> = 2'020'000'000'000'000'000;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...) \
  vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T floor(T a, T b) {
  return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
  return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sm = 0;
  for (auto &&a: A) sm += a;
  return sm;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    (check(x) ? ok : ng) = x;
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    (check(x) ? ok : ng) = x;
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}

template <typename T, typename... Vectors>
void concat(vc<T> &first, const Vectors &... others) {
  vc<T> &res = first;
  (res.insert(res.end(), others.begin(), others.end()), ...);
}
#endif
#line 1 "/home/maspy/compro/library/other/io.hpp"
#define FASTIO
#include <unistd.h>

// https://judge.yosupo.jp/submission/21623
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;

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

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

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

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

void rd(string &x) {
  x.clear();
  char c;
  do {
    if (pil + 1 > pir) load();
    c = ibuf[pil++];
  } while (isspace(c));
  do {
    x += c;
    if (pil == pir) load();
    c = ibuf[pil++];
  } while (!isspace(c));
}

template <typename T>
void rd_real(T &x) {
  string s;
  rd(s);
  x = stod(s);
}

template <typename T>
void rd_integer(T &x) {
  if (pil + 100 > pir) load();
  char c;
  do
    c = ibuf[pil++];
  while (c < '-');
  bool minus = 0;
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (c == '-') { minus = 1, c = ibuf[pil++]; }
  }
  x = 0;
  while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; }
  if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
    if (minus) x = -x;
  }
}

void rd(int &x) { rd_integer(x); }
void rd(ll &x) { rd_integer(x); }
void rd(i128 &x) { rd_integer(x); }
void rd(u32 &x) { rd_integer(x); }
void rd(u64 &x) { rd_integer(x); }
void rd(u128 &x) { rd_integer(x); }
void rd(double &x) { rd_real(x); }
void rd(long double &x) { rd_real(x); }
void rd(f128 &x) { rd_real(x); }

template <class T, class U>
void rd(pair<T, U> &p) {
  return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T>
void rd_tuple(T &t) {
  if constexpr (N < std::tuple_size<T>::value) {
    auto &x = std::get<N>(t);
    rd(x);
    rd_tuple<N + 1>(t);
  }
}
template <class... T>
void rd(tuple<T...> &tpl) {
  rd_tuple(tpl);
}

template <size_t N = 0, typename T>
void rd(array<T, N> &x) {
  for (auto &d: x) rd(d);
}
template <class T>
void rd(vc<T> &x) {
  for (auto &d: x) rd(d);
}

void read() {}
template <class H, class... T>
void read(H &h, T &... t) {
  rd(h), read(t...);
}

void wt(const char c) {
  if (por == SZ) flush();
  obuf[por++] = c;
}
void wt(const string s) {
  for (char c: s) wt(c);
}
void wt(const char *s) {
  size_t len = strlen(s);
  for (size_t i = 0; i < len; i++) wt(s[i]);
}

template <typename T>
void wt_integer(T x) {
  if (por > SZ - 100) flush();
  if (x < 0) { obuf[por++] = '-', x = -x; }
  int outi;
  for (outi = 96; x >= 10000; outi -= 4) {
    memcpy(out + outi, pre.num[x % 10000], 4);
    x /= 10000;
  }
  if (x >= 1000) {
    memcpy(obuf + por, pre.num[x], 4);
    por += 4;
  } else if (x >= 100) {
    memcpy(obuf + por, pre.num[x] + 1, 3);
    por += 3;
  } else if (x >= 10) {
    int q = (x * 103) >> 10;
    obuf[por] = q | '0';
    obuf[por + 1] = (x - q * 10) | '0';
    por += 2;
  } else
    obuf[por++] = x | '0';
  memcpy(obuf + por, out + outi + 4, 96 - outi);
  por += 96 - outi;
}

template <typename T>
void wt_real(T x) {
  ostringstream oss;
  oss << fixed << setprecision(15) << double(x);
  string s = oss.str();
  wt(s);
}

void wt(int x) { wt_integer(x); }
void wt(ll x) { wt_integer(x); }
void wt(i128 x) { wt_integer(x); }
void wt(u32 x) { wt_integer(x); }
void wt(u64 x) { wt_integer(x); }
void wt(u128 x) { wt_integer(x); }
void wt(double x) { wt_real(x); }
void wt(long double x) { wt_real(x); }
void wt(f128 x) { wt_real(x); }

template <class T, class U>
void wt(const pair<T, U> val) {
  wt(val.first);
  wt(' ');
  wt(val.second);
}
template <size_t N = 0, typename T>
void wt_tuple(const T t) {
  if constexpr (N < std::tuple_size<T>::value) {
    if constexpr (N > 0) { wt(' '); }
    const auto x = std::get<N>(t);
    wt(x);
    wt_tuple<N + 1>(t);
  }
}
template <class... T>
void wt(tuple<T...> tpl) {
  wt_tuple(tpl);
}
template <class T, size_t S>
void wt(const array<T, S> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}
template <class T>
void wt(const vector<T> val) {
  auto n = val.size();
  for (size_t i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(val[i]);
  }
}

void print() { wt('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt(head);
  if (sizeof...(Tail)) wt(' ');
  print(forward<Tail>(tail)...);
}

// gcc expansion. called automaticall after main.
void __attribute__((destructor)) _d() { flush(); }
} // namespace fastio
using fastio::read;
using fastio::print;
using fastio::flush;

#if defined(LOCAL)
#define SHOW(...) SHOW_IMPL(__VA_ARGS__, SHOW6, SHOW5, SHOW4, SHOW3, SHOW2, SHOW1)(__VA_ARGS__)
#define SHOW_IMPL(_1, _2, _3, _4, _5, _6, NAME, ...) NAME
#define SHOW1(x) print(#x, "=", (x)), flush()
#define SHOW2(x, y) print(#x, "=", (x), #y, "=", (y)), flush()
#define SHOW3(x, y, z) print(#x, "=", (x), #y, "=", (y), #z, "=", (z)), flush()
#define SHOW4(x, y, z, w) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w)), flush()
#define SHOW5(x, y, z, w, v) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v)), flush()
#define SHOW6(x, y, z, w, v, u) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v), #u, "=", (u)), flush()
#else
#define SHOW(...)
#endif

#define INT(...)   \
  int __VA_ARGS__; \
  read(__VA_ARGS__)
#define LL(...)   \
  ll __VA_ARGS__; \
  read(__VA_ARGS__)
#define U32(...)   \
  u32 __VA_ARGS__; \
  read(__VA_ARGS__)
#define U64(...)   \
  u64 __VA_ARGS__; \
  read(__VA_ARGS__)
#define STR(...)      \
  string __VA_ARGS__; \
  read(__VA_ARGS__)
#define CHAR(...)   \
  char __VA_ARGS__; \
  read(__VA_ARGS__)
#define DBL(...)      \
  double __VA_ARGS__; \
  read(__VA_ARGS__)

#define VEC(type, name, size) \
  vector<type> name(size);    \
  read(name)
#define VV(type, name, h, w)                     \
  vector<vector<type>> name(h, vector<type>(w)); \
  read(name)

void YES(bool t = 1) { print(t ? "YES" : "NO"); }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { print(t ? "Yes" : "No"); }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { print(t ? "yes" : "no"); }
void no(bool t = 1) { yes(!t); }
#line 3 "main.cpp"

#line 2 "/home/maspy/compro/library/graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  // sum(deg(v)) の計算量になっていて、
  // 新しいグラフの n+m より大きい可能性があるので注意
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (len(used_e) <= e.id) used_e.resize(e.id + 1);
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

  Graph<T, true> to_directed_tree(int root = -1) {
    if (root == -1) root = 0;
    assert(!is_directed && prepared && M == N - 1);
    Graph<T, true> G1(N);
    vc<int> par(N, -1);
    auto dfs = [&](auto& dfs, int v) -> void {
      for (auto& e: (*this)[v]) {
        if (e.to == par[v]) continue;
        par[e.to] = v, dfs(dfs, e.to);
      }
    };
    dfs(dfs, root);
    for (auto& e: edges) {
      int a = e.frm, b = e.to;
      if (par[a] == b) swap(a, b);
      assert(par[b] == a);
      G1.add(a, b, e.cost);
    }
    G1.build();
    return G1;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 2 "/home/maspy/compro/library/ds/unionfind/unionfind.hpp"

struct UnionFind {
  int n, n_comp;
  vc<int> dat; // par or (-size)
  UnionFind(int n = 0) { build(n); }

  void build(int m) {
    n = m, n_comp = m;
    dat.assign(n, -1);
  }

  void reset() { build(n); }

  int operator[](int x) {
    while (dat[x] >= 0) {
      int pp = dat[dat[x]];
      if (pp < 0) { return dat[x]; }
      x = dat[x] = pp;
    }
    return x;
  }

  ll size(int x) {
    x = (*this)[x];
    return -dat[x];
  }

  bool merge(int x, int y) {
    x = (*this)[x], y = (*this)[y];
    if (x == y) return false;
    if (-dat[x] < -dat[y]) swap(x, y);
    dat[x] += dat[y], dat[y] = x, n_comp--;
    return true;
  }

  vc<int> get_all() {
    vc<int> A(n);
    FOR(i, n) A[i] = (*this)[i];
    return A;
  }
};
#line 2 "/home/maspy/compro/library/mod/modint_common.hpp"

struct has_mod_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};

template <typename mint>
mint inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {0, 1};
  assert(0 <= n);
  if (n >= mod) n %= mod;
  while (len(dat) <= n) {
    int k = len(dat);
    int q = (mod + k - 1) / k;
    dat.eb(dat[k * q - mod] * mint::raw(q));
  }
  return dat[n];
}

template <typename mint>
mint fact(int n) {
  static const int mod = mint::get_mod();
  assert(0 <= n && n < mod);
  static vector<mint> dat = {1, 1};
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat)));
  return dat[n];
}

template <typename mint>
mint fact_inv(int n) {
  static vector<mint> dat = {1, 1};
  if (n < 0) return mint(0);
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
  return dat[n];
}

template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
  return (mint(1) * ... * fact_inv<mint>(xs));
}

template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
  return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}

template <typename mint>
mint C_dense(int n, int k) {
  static vvc<mint> C;
  static int H = 0, W = 0;
  auto calc = [&](int i, int j) -> mint {
    if (i == 0) return (j == 0 ? mint(1) : mint(0));
    return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
  };
  if (W <= k) {
    FOR(i, H) {
      C[i].resize(k + 1);
      FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
    }
    W = k + 1;
  }
  if (H <= n) {
    C.resize(n + 1);
    FOR(i, H, n + 1) {
      C[i].resize(W);
      FOR(j, W) { C[i][j] = calc(i, j); }
    }
    H = n + 1;
  }
  return C[n][k];
}

template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
  assert(n >= 0);
  if (k < 0 || n < k) return 0;
  if constexpr (dense) return C_dense<mint>(n, k);
  if constexpr (!large) return multinomial<mint>(n, k, n - k);
  k = min(k, n - k);
  mint x(1);
  FOR(i, k) x *= mint(n - i);
  return x * fact_inv<mint>(k);
}

template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
  assert(n >= 0);
  assert(0 <= k && k <= n);
  if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
  return mint(1) / C<mint, 1>(n, k);
}

// [x^d](1-x)^{-n}
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
  assert(n >= 0);
  if (d < 0) return mint(0);
  if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
  return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "/home/maspy/compro/library/mod/modint.hpp"

template <int mod>
struct modint {
  static constexpr u32 umod = u32(mod);
  static_assert(umod < u32(1) << 31);
  u32 val;

  static modint raw(u32 v) {
    modint x;
    x.val = v;
    return x;
  }
  constexpr modint() : val(0) {}
  constexpr modint(u32 x) : val(x % umod) {}
  constexpr modint(u64 x) : val(x % umod) {}
  constexpr modint(u128 x) : val(x % umod) {}
  constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
  bool operator<(const modint &other) const { return val < other.val; }
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += umod - p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = u64(val) * p.val % umod;
    return *this;
  }
  modint &operator/=(const modint &p) {
    *this *= p.inverse();
    return *this;
  }
  modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
  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 val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }
  modint inverse() const {
    int a = val, 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(ll n) const {
    assert(n >= 0);
    modint ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  static constexpr int get_mod() { return mod; }
  // (n, r), r は 1 の 2^n 乗根
  static constexpr pair<int, int> ntt_info() {
    if (mod == 120586241) return {20, 74066978};
    if (mod == 167772161) return {25, 17};
    if (mod == 469762049) return {26, 30};
    if (mod == 754974721) return {24, 362};
    if (mod == 880803841) return {23, 211};
    if (mod == 943718401) return {22, 663003469};
    if (mod == 998244353) return {23, 31};
    if (mod == 1004535809) return {21, 836905998};
    if (mod == 1045430273) return {20, 363};
    if (mod == 1051721729) return {20, 330};
    if (mod == 1053818881) return {20, 2789};
    return {-1, -1};
  }
  static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};

#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
  fastio::rd(x.val);
  x.val %= mod;
  // assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
  fastio::wt(x.val);
}
#endif

using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 2 "/home/maspy/compro/library/alg/monoid/add.hpp"

template <typename E>
struct Monoid_Add {
  using X = E;
  using value_type = X;
  static constexpr X op(const X &x, const X &y) noexcept { return x + y; }
  static constexpr X inverse(const X &x) noexcept { return -x; }
  static constexpr X power(const X &x, ll n) noexcept { return X(n) * x; }
  static constexpr X unit() { return X(0); }
  static constexpr bool commute = true;
};
#line 3 "/home/maspy/compro/library/ds/fenwicktree/fenwicktree.hpp"

template <typename Monoid>
struct FenwickTree {
  using G = Monoid;
  using MX = Monoid;
  using E = typename G::value_type;
  int n;
  vector<E> dat;
  E total;

  FenwickTree() {}
  FenwickTree(int n) { build(n); }
  template <typename F>
  FenwickTree(int n, F f) {
    build(n, f);
  }
  FenwickTree(const vc<E>& v) { build(v); }

  void build(int m) {
    n = m;
    dat.assign(m, G::unit());
    total = G::unit();
  }
  void build(const vc<E>& v) {
    build(len(v), [&](int i) -> E { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m;
    dat.clear();
    dat.reserve(n);
    total = G::unit();
    FOR(i, n) { dat.eb(f(i)); }
    for (int i = 1; i <= n; ++i) {
      int j = i + (i & -i);
      if (j <= n) dat[j - 1] = G::op(dat[i - 1], dat[j - 1]);
    }
    total = prefix_sum(m);
  }

  E prod_all() { return total; }
  E sum_all() { return total; }
  E sum(int k) { return prefix_sum(k); }
  E prod(int k) { return prefix_prod(k); }
  E prefix_sum(int k) { return prefix_prod(k); }
  E prefix_prod(int k) {
    chmin(k, n);
    E ret = G::unit();
    for (; k > 0; k -= k & -k) ret = G::op(ret, dat[k - 1]);
    return ret;
  }
  E sum(int L, int R) { return prod(L, R); }
  E prod(int L, int R) {
    chmax(L, 0), chmin(R, n);
    if (L == 0) return prefix_prod(R);
    assert(0 <= L && L <= R && R <= n);
    E pos = G::unit(), neg = G::unit();
    while (L < R) { pos = G::op(pos, dat[R - 1]), R -= R & -R; }
    while (R < L) { neg = G::op(neg, dat[L - 1]), L -= L & -L; }
    return G::op(pos, G::inverse(neg));
  }

  vc<E> get_all() {
    vc<E> res(n);
    FOR(i, n) res[i] = prod(i, i + 1);
    return res;
  }

  void add(int k, E x) { multiply(k, x); }
  void multiply(int k, E x) {
    static_assert(G::commute);
    total = G::op(total, x);
    for (++k; k <= n; k += k & -k) dat[k - 1] = G::op(dat[k - 1], x);
  }

  template <class F>
  int max_right(const F check, int L = 0) {
    assert(check(G::unit()));
    E s = G::unit();
    int i = L;
    // 2^k 進むとダメ
    int k = [&]() {
      while (1) {
        if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
        if (i == 0) { return topbit(n) + 1; }
        int k = lowbit(i) - 1;
        if (i + (1 << k) > n) return k;
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (!check(t)) { return k; }
        s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
      }
    }();
    while (k) {
      --k;
      if (i + (1 << k) - 1 < len(dat)) {
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (check(t)) { i += (1 << k), s = t; }
      }
    }
    return i;
  }

  // check(i, x)
  template <class F>
  int max_right_with_index(const F check, int L = 0) {
    assert(check(L, G::unit()));
    E s = G::unit();
    int i = L;
    // 2^k 進むとダメ
    int k = [&]() {
      while (1) {
        if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
        if (i == 0) { return topbit(n) + 1; }
        int k = lowbit(i) - 1;
        if (i + (1 << k) > n) return k;
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (!check(i + (1 << k), t)) { return k; }
        s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
      }
    }();
    while (k) {
      --k;
      if (i + (1 << k) - 1 < len(dat)) {
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (check(i + (1 << k), t)) { i += (1 << k), s = t; }
      }
    }
    return i;
  }

  template <class F>
  int min_left(const F check, int R) {
    assert(check(G::unit()));
    E s = G::unit();
    int i = R;
    // false になるところまで戻る
    int k = 0;
    while (i > 0 && check(s)) {
      s = G::op(s, dat[i - 1]);
      k = lowbit(i);
      i -= i & -i;
    }
    if (check(s)) {
      assert(i == 0);
      return 0;
    }
    // 2^k 進むと ok になる
    // false を維持して進む
    while (k) {
      --k;
      E t = G::op(s, G::inverse(dat[i + (1 << k) - 1]));
      if (!check(t)) { i += (1 << k), s = t; }
    }
    return i + 1;
  }

  int kth(E k, int L = 0) {
    return max_right([&k](E x) -> bool { return x <= k; }, L);
  }
};
#line 2 "/home/maspy/compro/library/graph/tree.hpp"

#line 4 "/home/maspy/compro/library/graph/tree.hpp"

// HLD euler tour をとっていろいろ。
template <typename GT>
struct Tree {
  using Graph_type = GT;
  GT &G;
  using WT = typename GT::cost_type;
  int N;
  vector<int> LID, RID, head, V, parent, VtoE;
  vc<int> depth;
  vc<WT> depth_weighted;

  Tree(GT &G, int r = 0, bool hld = 1) : G(G) { build(r, hld); }

  void build(int r = 0, bool hld = 1) {
    if (r == -1) return; // build を遅延したいとき
    N = G.N;
    LID.assign(N, -1), RID.assign(N, -1), head.assign(N, r);
    V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1);
    depth.assign(N, -1), depth_weighted.assign(N, 0);
    assert(G.is_prepared());
    int t1 = 0;
    dfs_sz(r, -1, hld);
    dfs_hld(r, t1);
  }

  void dfs_sz(int v, int p, bool hld) {
    auto &sz = RID;
    parent[v] = p;
    depth[v] = (p == -1 ? 0 : depth[p] + 1);
    sz[v] = 1;
    int l = G.indptr[v], r = G.indptr[v + 1];
    auto &csr = G.csr_edges;
    // 使う辺があれば先頭にする
    for (int i = r - 2; i >= l; --i) {
      if (hld && depth[csr[i + 1].to] == -1) swap(csr[i], csr[i + 1]);
    }
    int hld_sz = 0;
    for (int i = l; i < r; ++i) {
      auto e = csr[i];
      if (depth[e.to] != -1) continue;
      depth_weighted[e.to] = depth_weighted[v] + e.cost;
      VtoE[e.to] = e.id;
      dfs_sz(e.to, v, hld);
      sz[v] += sz[e.to];
      if (hld && chmax(hld_sz, sz[e.to]) && l < i) { swap(csr[l], csr[i]); }
    }
  }

  void dfs_hld(int v, int &times) {
    LID[v] = times++;
    RID[v] += LID[v];
    V[LID[v]] = v;
    bool heavy = true;
    for (auto &&e: G[v]) {
      if (depth[e.to] <= depth[v]) continue;
      head[e.to] = (heavy ? head[v] : e.to);
      heavy = false;
      dfs_hld(e.to, times);
    }
  }

  vc<int> heavy_path_at(int v) {
    vc<int> P = {v};
    while (1) {
      int a = P.back();
      for (auto &&e: G[a]) {
        if (e.to != parent[a] && head[e.to] == v) {
          P.eb(e.to);
          break;
        }
      }
      if (P.back() == a) break;
    }
    return P;
  }

  int heavy_child(int v) {
    int k = LID[v] + 1;
    if (k == N) return -1;
    int w = V[k];
    return (parent[w] == v ? w : -1);
  }

  int e_to_v(int eid) {
    auto e = G.edges[eid];
    return (parent[e.frm] == e.to ? e.frm : e.to);
  }
  int v_to_e(int v) { return VtoE[v]; }
  int get_eid(int u, int v) {
    if (parent[u] != v) swap(u, v);
    assert(parent[u] == v);
    return VtoE[u];
  }

  int ELID(int v) { return 2 * LID[v] - depth[v]; }
  int ERID(int v) { return 2 * RID[v] - depth[v] - 1; }

  // 目標地点へ進む個数が k
  int LA(int v, int k) {
    assert(k <= depth[v]);
    while (1) {
      int u = head[v];
      if (LID[v] - k >= LID[u]) return V[LID[v] - k];
      k -= LID[v] - LID[u] + 1;
      v = parent[u];
    }
  }
  int la(int u, int v) { return LA(u, v); }

  int LCA(int u, int v) {
    for (;; v = parent[head[v]]) {
      if (LID[u] > LID[v]) swap(u, v);
      if (head[u] == head[v]) return u;
    }
  }

  int meet(int a, int b, int c) { return LCA(a, b) ^ LCA(a, c) ^ LCA(b, c); }
  int lca(int u, int v) { return LCA(u, v); }

  int subtree_size(int v, int root = -1) {
    if (root == -1) return RID[v] - LID[v];
    if (v == root) return N;
    int x = jump(v, root, 1);
    if (in_subtree(v, x)) return RID[v] - LID[v];
    return N - RID[x] + LID[x];
  }

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

  WT dist_weighted(int a, int b) {
    int c = LCA(a, b);
    return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c];
  }

  // a is in b
  bool in_subtree(int a, int b) { return LID[b] <= LID[a] && LID[a] < RID[b]; }

  int jump(int a, int b, ll k) {
    if (k == 1) {
      if (a == b) return -1;
      return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]);
    }
    int c = LCA(a, b);
    int d_ac = depth[a] - depth[c];
    int d_bc = depth[b] - depth[c];
    if (k > d_ac + d_bc) return -1;
    if (k <= d_ac) return LA(a, k);
    return LA(b, d_ac + d_bc - k);
  }

  vc<int> collect_child(int v) {
    vc<int> res;
    for (auto &&e: G[v])
      if (e.to != parent[v]) res.eb(e.to);
    return res;
  }

  vc<int> collect_light(int v) {
    vc<int> res;
    bool skip = true;
    for (auto &&e: G[v])
      if (e.to != parent[v]) {
        if (!skip) res.eb(e.to);
        skip = false;
      }
    return res;
  }

  vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) {
    // [始点, 終点] の"閉"区間列。
    vc<pair<int, int>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        down.eb(LID[head[v]], LID[v]);
        v = parent[head[v]];
      } else {
        up.eb(LID[u], LID[head[u]]);
        u = parent[head[u]];
      }
    }
    if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]);
    elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge);
    reverse(all(down));
    up.insert(up.end(), all(down));
    return up;
  }

  // 辺の列の情報 (frm,to,str)
  // str = "heavy_up", "heavy_down", "light_up", "light_down"
  vc<tuple<int, int, string>> get_path_decomposition_detail(int u, int v) {
    vc<tuple<int, int, string>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        if (v != head[v]) down.eb(head[v], v, "heavy_down"), v = head[v];
        down.eb(parent[v], v, "light_down"), v = parent[v];
      } else {
        if (u != head[u]) up.eb(u, head[u], "heavy_up"), u = head[u];
        up.eb(u, parent[u], "light_up"), u = parent[u];
      }
    }
    if (LID[u] < LID[v]) down.eb(u, v, "heavy_down");
    elif (LID[v] < LID[u]) up.eb(u, v, "heavy_up");
    reverse(all(down));
    concat(up, down);
    return up;
  }

  vc<int> restore_path(int u, int v) {
    vc<int> P;
    for (auto &&[a, b]: get_path_decomposition(u, v, 0)) {
      if (a <= b) {
        FOR(i, a, b + 1) P.eb(V[i]);
      } else {
        FOR_R(i, b, a + 1) P.eb(V[i]);
      }
    }
    return P;
  }

  // path [a,b] と [c,d] の交わり. 空ならば {-1,-1}.
  // https://codeforces.com/problemset/problem/500/G
  pair<int, int> path_intersection(int a, int b, int c, int d) {
    int ab = lca(a, b), ac = lca(a, c), ad = lca(a, d);
    int bc = lca(b, c), bd = lca(b, d), cd = lca(c, d);
    int x = ab ^ ac ^ bc, y = ab ^ ad ^ bd; // meet(a,b,c), meet(a,b,d)
    if (x != y) return {x, y};
    int z = ac ^ ad ^ cd;
    if (x != z) x = -1;
    return {x, x};
  }
};
#line 3 "/home/maspy/compro/library/graph/ds/tree_abelgroup.hpp"

template <typename TREE, typename AbelGroup, bool edge, bool path_query,
          bool subtree_query>
struct Tree_AbelGroup {
  using MX = AbelGroup;
  using X = typename MX::value_type;
  TREE &tree;
  int N;
  FenwickTree<MX> bit, bit_subtree;

  Tree_AbelGroup(TREE &tree) : tree(tree), N(tree.N) {
    build([](int i) -> X { return MX::unit(); });
  }

  Tree_AbelGroup(TREE &tree, vc<X> &dat) : tree(tree), N(tree.N) {
    build([&](int i) -> X { return dat[i]; });
  }

  template <typename F>
  Tree_AbelGroup(TREE &tree, F f) : tree(tree), N(tree.N) {
    build(f);
  }

  template <typename F>
  void build(F f) {
    vc<X> bit_raw_1(2 * N);
    vc<X> bit_raw_2(N);
    FOR(v, N) {
      X x = MX::unit();
      if (!edge) x = f(v);
      if (edge) x = (v == 0 ? MX::unit() : f(tree.v_to_e(v)));
      bit_raw_1[tree.ELID(v)] = x;
      bit_raw_1[tree.ERID(v)] = MX::inverse(x);
      bit_raw_2[tree.LID[v]] = x;
    }
    if constexpr (path_query) bit.build(bit_raw_1);
    if constexpr (subtree_query) bit_subtree.build(bit_raw_2);
  }

  void add(int i, X x) {
    int v = (edge ? tree.e_to_v(i) : i);
    if constexpr (path_query) {
      bit.add(tree.ELID(v), x);
      bit.add(tree.ERID(v), MX::inverse(x));
    }
    if constexpr (subtree_query) bit_subtree.add(tree.LID[v], x);
  }

  X prod_path(int frm, int to) {
    static_assert(path_query);
    int lca = tree.LCA(frm, to);
    // [frm, lca)
    X x1 = bit.prod(tree.ELID(lca) + 1, tree.ELID(frm) + 1);
    // edge なら (lca, to]、vertex なら [lca, to]
    X x2 = bit.prod(tree.ELID(lca) + edge, tree.ELID(to) + 1);
    return MX::op(x1, x2);
  }

  X prod_subtree(int u, int root = -1) {
    static_assert(subtree_query);
    int l = tree.LID[u], r = tree.RID[u];
    if (root == -1) return bit_subtree.prod(l + edge, r);
    if (root == u) return bit_subtree.prod_all();
    if (tree.in_subtree(u, root)) return bit_subtree.prod(l + edge, r);
    return MX::op(bit_subtree.prod(0, l + 1), bit_subtree.prod(r, N));
  }
};
#line 2 "/home/maspy/compro/library/graph/compress_tree.hpp"

// (圧縮された木の頂点ラベルたち、グラフ)
// 新しいグラフ：辺重みあり
template <typename TREE>
pair<vc<int>, typename TREE::Graph_type> compress_tree(TREE& tree, vc<int> V) {
  // 大事な点をリストアップする
  sort(all(V), [&](auto& x, auto& y) { return tree.LID[x] < tree.LID[y]; });
  int n = len(V);
  FOR(i, n) {
    int j = (i + 1 == n ? 0 : i + 1);
    V.eb(tree.lca(V[i], V[j]));
  }
  sort(all(V), [&](auto& x, auto& y) { return tree.LID[x] < tree.LID[y]; });
  V.erase(unique(all(V)), V.end());
  // 辺を張ってグラフを作る
  n = len(V);
  using GT = typename TREE::Graph_type;
  using WT = typename GT::cost_type;
  GT G(n);
  vc<int> st = {0};
  FOR(i, 1, n) {
    while (1) {
      int p = V[st.back()];
      int v = V[i];
      if (tree.in_subtree(v, p)) break;
      st.pop_back();
    }
    int p = V[st.back()];
    int v = V[i];
    WT d = tree.depth_weighted[v] - tree.depth_weighted[p];
    G.add(st.back(), i, d);
    st.eb(i);
  }
  G.build();
  return {V, G};
}
#line 9 "main.cpp"

using mint = modint107;

void solve() {
  LL(N, M);

  Graph<int, 0> G(N);
  G.read_graph(M);

  Graph<int, 1> H(N);
  UnionFind uf(N);
  vc<int> root(N);
  FOR(i, N) root[i] = i;
  FOR(v, N) {
    for (auto& e: G[v]) {
      int w = e.to;
      if (uf[v] == uf[w]) continue;
      if (v < w) continue;
      int a = uf[v], b = uf[w];
      H.add(v, root[b]);
      uf.merge(a, b);
      root[uf[a]] = v;
    }
  }
  H.build();

  // H.debug();

  Tree<decltype(H)> tree(H, N - 1);
  Tree_AbelGroup<decltype(tree), Monoid_Add<mint>, false, true, false> TA(tree);
  vc<mint> dp(N);

  FOR(v, N) {
    vc<int> V;
    V.eb(v);
    for (auto& e: G[v]) {
      if (e.to < v) V.eb(e.to);
    }
    sort(all(V), [&](auto& L, auto& R) -> bool { return tree.LID[L] < tree.LID[R]; });
    mint ans = 1;
    FOR(i, 1, len(V)) {
      int v = V[i];
      int p = tree.lca(V[i - 1], v);
      p = tree.jump(p, v, 1);
      ans += TA.prod_path(p, v);
    }
    dp[v] = ans;
    TA.add(v, ans);
  }

  mint ANS = SUM<mint>(dp);
  print(ANS);
}

signed main() {
  int T = 1;
  // INT(T);
  FOR(T) solve();
  return 0;
}