/**
 * date   : 2024-12-10 15:09:59
 * author : Nyaan
 */

#define NDEBUG

using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tr2/dynamic_bitset>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// utility

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

}  // namespace Nyaan


// bit operation

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


// inout

namespace Nyaan {

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

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

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

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

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

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

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

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

}  // namespace Nyaan


// debug


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

#ifndef NyaanLocal
#define trc2(...) (void(0))
#endif


// macro

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


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


//


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([[maybe_unused]] 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
 */


//


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; }
};





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(long long n, long long r) {
    if (n < 0 || r < 0) return T(0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};


//

namespace nachia {

template <class Elem>
class CsrArray {
 public:
  struct ListRange {
    using iterator = typename std::vector<Elem>::iterator;
    iterator begi, endi;
    iterator begin() const { return begi; }
    iterator end() const { return endi; }
    int size() const { return (int)std::distance(begi, endi); }
    Elem& operator[](int i) const { return begi[i]; }
  };
  struct ConstListRange {
    using iterator = typename std::vector<Elem>::const_iterator;
    iterator begi, endi;
    iterator begin() const { return begi; }
    iterator end() const { return endi; }
    int size() const { return (int)std::distance(begi, endi); }
    const Elem& operator[](int i) const { return begi[i]; }
  };

 private:
  int m_n;
  std::vector<Elem> m_list;
  std::vector<int> m_pos;

 public:
  CsrArray() : m_n(0), m_list(), m_pos() {}
  static CsrArray Construct(int n, std::vector<std::pair<int, Elem>> items) {
    CsrArray res;
    res.m_n = n;
    std::vector<int> buf(n + 1, 0);
    for (auto& [u, v] : items) {
      ++buf[u];
    }
    for (int i = 1; i <= n; i++) buf[i] += buf[i - 1];
    res.m_list.resize(buf[n]);
    for (int i = (int)items.size() - 1; i >= 0; i--) {
      res.m_list[--buf[items[i].first]] = std::move(items[i].second);
    }
    res.m_pos = std::move(buf);
    return res;
  }
  static CsrArray FromRaw(std::vector<Elem> list, std::vector<int> pos) {
    CsrArray res;
    res.m_n = pos.size() - 1;
    res.m_list = std::move(list);
    res.m_pos = std::move(pos);
    return res;
  }
  ListRange operator[](int u) {
    return ListRange{m_list.begin() + m_pos[u], m_list.begin() + m_pos[u + 1]};
  }
  ConstListRange operator[](int u) const {
    return ConstListRange{m_list.begin() + m_pos[u],
                          m_list.begin() + m_pos[u + 1]};
  }
  int size() const { return m_n; }
  int fullSize() const { return (int)m_list.size(); }
};

}  // namespace nachia

namespace nachia {

struct Graph {
 public:
  struct Edge {
    int from, to;
    void reverse() { std::swap(from, to); }
    int xorval() const { return from ^ to; }
  };
  Graph(int n = 0, bool undirected = false, int m = 0)
      : m_n(n), m_e(m), m_isUndir(undirected) {}
  Graph(int n, const std::vector<std::pair<int, int>>& edges,
        bool undirected = false)
      : m_n(n), m_isUndir(undirected) {
    m_e.resize(edges.size());
    for (std::size_t i = 0; i < edges.size(); i++)
      m_e[i] = {edges[i].first, edges[i].second};
  }
  template <class Cin>
  static Graph Input(Cin& cin, int n, bool undirected, int m, bool offset = 0) {
    Graph res(n, undirected, m);
    for (int i = 0; i < m; i++) {
      int u, v;
      cin >> u >> v;
      res[i].from = u - offset;
      res[i].to = v - offset;
    }
    return res;
  }
  int numVertices() const noexcept { return m_n; }
  int numEdges() const noexcept { return int(m_e.size()); }
  int addNode() noexcept { return m_n++; }
  int addEdge(int from, int to) {
    m_e.push_back({from, to});
    return numEdges() - 1;
  }
  Edge& operator[](int ei) noexcept { return m_e[ei]; }
  const Edge& operator[](int ei) const noexcept { return m_e[ei]; }
  Edge& at(int ei) { return m_e.at(ei); }
  const Edge& at(int ei) const { return m_e.at(ei); }
  auto begin() { return m_e.begin(); }
  auto end() { return m_e.end(); }
  auto begin() const { return m_e.begin(); }
  auto end() const { return m_e.end(); }
  bool isUndirected() const noexcept { return m_isUndir; }
  void reverseEdges() noexcept {
    for (auto& e : m_e) e.reverse();
  }
  void contract(int newV, const std::vector<int>& mapping) {
    assert(numVertices() == int(mapping.size()));
    for (int i = 0; i < numVertices(); i++)
      assert(0 <= mapping[i] && mapping[i] < newV);
    for (auto& e : m_e) {
      e.from = mapping[e.from];
      e.to = mapping[e.to];
    }
    m_n = newV;
  }
  std::vector<Graph> induce(int num, const std::vector<int>& mapping) const {
    int n = numVertices();
    assert(n == int(mapping.size()));
    for (int i = 0; i < n; i++) assert(-1 <= mapping[i] && mapping[i] < num);
    std::vector<int> indexV(n), newV(num);
    for (int i = 0; i < n; i++)
      if (mapping[i] >= 0) indexV[i] = newV[mapping[i]]++;
    std::vector<Graph> res;
    res.reserve(num);
    for (int i = 0; i < num; i++) res.emplace_back(newV[i], isUndirected());
    for (auto e : m_e)
      if (mapping[e.from] == mapping[e.to] && mapping[e.to] >= 0)
        res[mapping[e.to]].addEdge(indexV[e.from], indexV[e.to]);
    return res;
  }
  CsrArray<int> getEdgeIndexArray(bool undirected) const {
    std::vector<std::pair<int, int>> src;
    src.reserve(numEdges() * (undirected ? 2 : 1));
    for (int i = 0; i < numEdges(); i++) {
      auto e = operator[](i);
      src.emplace_back(e.from, i);
      if (undirected) src.emplace_back(e.to, i);
    }
    return CsrArray<int>::Construct(numVertices(), src);
  }
  CsrArray<int> getEdgeIndexArray() const {
    return getEdgeIndexArray(isUndirected());
  }
  CsrArray<int> getAdjacencyArray(bool undirected) const {
    std::vector<std::pair<int, int>> src;
    src.reserve(numEdges() * (undirected ? 2 : 1));
    for (auto e : m_e) {
      src.emplace_back(e.from, e.to);
      if (undirected) src.emplace_back(e.to, e.from);
    }
    return CsrArray<int>::Construct(numVertices(), src);
  }
  CsrArray<int> getAdjacencyArray() const {
    return getAdjacencyArray(isUndirected());
  }

 private:
  int m_n;
  std::vector<Edge> m_e;
  bool m_isUndir;
};

}  // namespace nachia

namespace nachia {

// simple graph
// for each edge
// O( n + m sqrt(m) ) time
template <class Weight>
std::vector<long long> CountC4Simple(int n, std::vector<int> U,
                                     std::vector<int> V,
                                     std::vector<Weight> W) {
  int m = int(W.size());

  // less incident edges, smaller index
  std::vector<int> deg(n);
  for (int e = 0; e < m; e++) {
    deg[U[e]]++;
    deg[V[e]]++;
  }
  std::vector<int> I(n);
  for (int i = 0; i < n; i++) I[i] = i;
  std::sort(I.begin(), I.end(), [&](int l, int r) { return deg[l] < deg[r]; });
  {
    std::vector<int> O(n);
    for (int i = 0; i < n; i++) O[I[i]] = i;
    for (int& u : U) u = O[u];
    for (int& u : V) u = O[u];
  }

  for (int e = 0; e < m; e++)
    if (U[e] < V[e]) std::swap(U[e], V[e]);

  // adjacency list

  std::vector<int> estart(n);
  for (int i = 0; i < n - 1; i++) estart[i + 1] = estart[i] + deg[I[i]];
  std::vector<int> eend = estart;
  std::vector<int> eid(m * 2);
  std::vector<int> eto(m * 2);

  for (int e = 0; e < m; e++) {
    int v = U[e];
    int w = V[e];
    eid[eend[v]] = e;
    eto[eend[v]] = w;
    eend[v]++;
  }
  std::vector<int> eendx = eend;
  for (int v = 0; v < n; v++) {
    for (int i = estart[v]; i < eendx[v]; i++) {
      int e = eid[i];
      int w = eto[i];
      eid[eend[w]] = e;
      eto[eend[w]] = v;
      eend[w]++;
    }
  }

  std::vector<Weight> c(n);  // c[x] : number of paths(v --> w --> x)
  std::vector<Weight> ans(m);
  for (int v = n - 1; v >= 0; v--) {
    for (int i = estart[v]; i < eend[v]; i++) {
      int evw = eid[i];
      int w = eto[i];
      eend[w] -= 1;  // remove w -> v
      for (int j = estart[w]; j < eend[w]; j++) {
        int ewx = eid[j];
        int x = eto[j];
        c[x] += W[evw] * W[ewx];
      }
    }
    for (int i = estart[v]; i < eend[v]; i++) {
      int evw = eid[i];
      int w = eto[i];
      for (int j = estart[w]; j < eend[w]; j++) {
        int ewx = eid[j];
        int x = eto[j];
        Weight val = c[x] - W[evw] * W[ewx];
        ans[evw] += val * W[ewx];
        ans[ewx] += val * W[evw];
      }
    }
    for (int i = estart[v]; i < eend[v]; i++) {
      int w = eto[i];
      for (int j = estart[w]; j < eend[w]; j++) c[eto[j]] = 0;
    }
  }
  return ans;
}

// for each edge
// O( n + m sqrt(m) ) time
template <class Weight>
std::vector<Weight> CountC4(int n, std::vector<int> U, std::vector<int> V,
                            std::vector<Weight> W) {
  int m = int(W.size());
  for (int i = 0; i < m; i++)
    if (U[i] > V[i]) std::swap(U[i], V[i]);
  std::vector<int> I(m);
  for (int i = 0; i < m; i++) I[i] = i;
  std::sort(I.begin(), I.end(), [&](int l, int r) { return V[l] < V[r]; });
  std::stable_sort(I.begin(), I.end(),
                   [&](int l, int r) { return U[l] < U[r]; });

  std::vector<int> Q(m);
  std::vector<int> U2;
  std::vector<int> V2;
  std::vector<Weight> W2;
  for (int i = 0; i < m; i++) {
    int e = I[i];
    if (i == 0 || U2.back() != U[e] || V2.back() != V[e]) {
      U2.push_back(U[e]);
      V2.push_back(V[e]);
      W2.push_back(0);
    }
    W2.back() += W[e];
    Q[e] = int(U2.size()) - 1;
  }

  auto simple_res =
      CountC4Simple<Weight>(n, std::move(U2), std::move(V2), std::move(W2));
  std::vector<Weight> ans(m);
  for (int e = 0; e < m; e++) ans[e] = simple_res[Q[e]];
  return ans;
}

}  // namespace nachia

//
using namespace Nyaan;
using mint = LazyMontgomeryModInt<998244353>;
// using mint = LazyMontgomeryModInt<1000000007>;
using vm = vector<mint>;
using vvm = vector<vm>;

using namespace Nyaan;

template <typename F>
void enumerate_triangle(const vvi& g, const F& f) {
  int N = sz(g);
  auto ord = mkord(N, [&](int i, int j) { return g[i].size() < g[j].size(); });
  auto inv = mkinv(ord);
  vvi h(N);
  vp es;
  rep(i, N) each(j, g[i]) if (inv[i] < inv[j]) {
    es.emplace_back(i, j), h[i].push_back(j);
  }
  V<bool> flg(N, 0);
  each(e, es) {
    each(u, h[e.first]) flg[u] = 1;
    each(v, h[e.second]) if (flg[v]) f(v, e.first, e.second);
    each(u, h[e.first]) flg[u] = 0;
  }
}

void q() {
  ini(N, M);
  vvi g = graph(N, M);
  vp es;
  rep(i, N) each(j, g[i]) if (i < j) es.emplace_back(i, j);
  vi deg(N);
  rep(i, N) deg[i] = sz(g[i]);

  ll C3_num = 0;
  enumerate_triangle(g, [&](int, int, int) { C3_num++; });

  ll C4_num = 0;
  {
    vi u, v;
    each2(i, j, es) u.push_back(i), v.push_back(j);
    auto c4 = nachia::CountC4(N, u, v, vl(M, 1));
    C4_num = Sum(c4) / 4;
  }

  Binomial<mint> C;
  auto f1 = [&]() -> mint { return C(N, 4); };
  auto f2 = [&]() -> mint { return C(N - 2, 2) * M; };
  auto f3 = [&]() -> mint {
    mint s = 0;
    rep(i, N) s += C(deg[i], 2);
    return s * (N - 3);
  };
  auto f4 = [&]() -> mint {
    mint s = 0;
    rep(i, N) s += C(deg[i], 2);
    return C(M, 2) - s;
  };
  auto f5 = [&]() -> mint {
    mint s = 0;
    rep(i, N) s += C(deg[i], 3);
    return s;
  };
  auto f6 = [&]() -> mint {
    mint s = 0;
    each2(u, v, es) s += (deg[u] - 1) * (deg[v] - 1);
    s -= 3 * C3_num;
    return s;
  };
  auto f7 = [&]() -> mint { return mint{C3_num} * (N - 3); };
  auto f8 = [&]() -> mint { return C4_num; };
  auto f9 = [&]() -> mint {
    vi cnt(N);
    enumerate_triangle(
        g, [&](int i, int j, int k) { cnt[i]++, cnt[j]++, cnt[k]++; });
    mint s = 0;
    rep(u, N) s += 1LL * (deg[u] - 2) * cnt[u];
    return s;
  };
  auto f10 = [&]() -> mint {
    vl v;
    enumerate_triangle(g, [&](ll i, ll j, ll k) {
      v.push_back((min(i, j) << 32) + max(i, j));
      v.push_back((min(j, k) << 32) + max(j, k));
      v.push_back((min(k, i) << 32) + max(k, i));
    });
    sort(begin(v), end(v));
    mint s = 0;
    for (int i = 0, j = 0; i < sz(v); i = j) {
      while (j != sz(v) and v[i] == v[j]) j++;
      s += C(j - i, 2);
    }
    return s;
  };
  trc(f1(), f2(), f3(), f4(), f5(), f6(), f7(), f8(), f9(), f10());
  mint T = mint{-1} / 3;
  mint ans = 0;
  ans += f1();
  ans -= f2();
  ans += f3();
  ans += f4();
  ans -= f5();
  ans -= f6();
  ans -= f7();
  ans += f8() * 2 / 3;
  ans += f9();
  ans -= f10() * 2 / 3;
  out(T, ans);
  // trc(mint{13} + T * 2);
}

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