/**
 * date   : 2023-12-21 22:07:15
 * author : Nyaan
 */

#define NDEBUG

using namespace std;

// intrinstic
#include <immintrin.h>

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

// utility

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

}  // namespace Nyaan


// bit operation

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


// inout

namespace Nyaan {

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

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

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

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

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

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

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

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

}  // namespace Nyaan


// debug


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

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


// macro

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


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


//


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
 */


//


template <typename T>
struct BinaryIndexedTree {
  int N;
  vector<T> data;

  BinaryIndexedTree() = default;

  BinaryIndexedTree(int size) { init(size); }

  void init(int size) {
    N = size + 2;
    data.assign(N + 1, {});
  }

  // get sum of [0,k]
  T sum(int k) const {
    if (k < 0) return T{};  // return 0 if k < 0
    T ret{};
    for (++k; k > 0; k -= k & -k) ret += data[k];
    return ret;
  }

  // getsum of [l,r]
  inline T sum(int l, int r) const { return sum(r) - sum(l - 1); }

  // get value of k
  inline T operator[](int k) const { return sum(k) - sum(k - 1); }

  // data[k] += x
  void add(int k, T x) {
    for (++k; k < N; k += k & -k) data[k] += x;
  }

  // range add x to [l,r]
  void imos(int l, int r, T x) {
    add(l, x);
    add(r + 1, -x);
  }

  // minimize i s.t. sum(i) >= w
  int lower_bound(T w) {
    if (w <= 0) return 0;
    int x = 0;
    for (int k = 1 << __lg(N); k; k >>= 1) {
      if (x + k <= N - 1 && data[x + k] < w) {
        w -= data[x + k];
        x += k;
      }
    }
    return x;
  }

  // minimize i s.t. sum(i) > w
  int upper_bound(T w) {
    if (w < 0) return 0;
    int x = 0;
    for (int k = 1 << __lg(N); k; k >>= 1) {
      if (x + k <= N - 1 && data[x + k] <= w) {
        w -= data[x + k];
        x += k;
      }
    }
    return x;
  }
};

/**
 * @brief Binary Indexed Tree(Fenwick Tree)
 * @docs docs/data-structure/binary-indexed-tree.md
 */




struct UnionFind {
  vector<int> data;
  UnionFind(int N) : data(N, -1) {}

  int find(int k) { return data[k] < 0 ? k : data[k] = find(data[k]); }

  int unite(int x, int y) {
    if ((x = find(x)) == (y = find(y))) return false;
    if (data[x] > data[y]) swap(x, y);
    data[x] += data[y];
    data[y] = x;
    return true;
  }

  // f ... merge function
  template<typename F>
  int unite(int x, int y,const F &f) {
    if ((x = find(x)) == (y = find(y))) return false;
    if (data[x] > data[y]) swap(x, y);
    data[x] += data[y];
    data[y] = x;
    f(x, y);
    return true;
  }

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

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

/**
 * @brief Union Find(Disjoint Set Union)
 * @docs docs/data-structure/union-find.md
 */




template <typename T, typename F>
struct SegmentTree {
  int N;
  int size;
  vector<T> seg;
  const F f;
  const T I;

  SegmentTree(F _f, const T &I_) : N(0), size(0), f(_f), I(I_) {}

  SegmentTree(int _N, F _f, const T &I_) : f(_f), I(I_) { init(_N); }

  SegmentTree(const vector<T> &v, F _f, T I_) : f(_f), I(I_) {
    init(v.size());
    for (int i = 0; i < (int)v.size(); i++) {
      seg[i + size] = v[i];
    }
    build();
  }

  void init(int _N) {
    N = _N;
    size = 1;
    while (size < N) size <<= 1;
    seg.assign(2 * size, I);
  }

  void set(int k, T x) { seg[k + size] = x; }

  void build() {
    for (int k = size - 1; k > 0; k--) {
      seg[k] = f(seg[2 * k], seg[2 * k + 1]);
    }
  }

  void update(int k, T x) {
    k += size;
    seg[k] = x;
    while (k >>= 1) {
      seg[k] = f(seg[2 * k], seg[2 * k + 1]);
    }
  }

  void add(int k, T x) {
    k += size;
    seg[k] += x;
    while (k >>= 1) {
      seg[k] = f(seg[2 * k], seg[2 * k + 1]);
    }
  }

  // query to [a, b)
  T query(int a, int b) {
    T L = I, R = I;
    for (a += size, b += size; a < b; a >>= 1, b >>= 1) {
      if (a & 1) L = f(L, seg[a++]);
      if (b & 1) R = f(seg[--b], R);
    }
    return f(L, R);
  }

  T &operator[](const int &k) { return seg[k + size]; }

  // check(a[l] * ...  * a[r-1]) が true となる最大の r
  // (右端まですべて true なら N を返す)
  template <class C>
  int max_right(int l, C check) {
    assert(0 <= l && l <= N);
    assert(check(I) == true);
    if (l == N) return N;
    l += size;
    T sm = I;
    do {
      while (l % 2 == 0) l >>= 1;
      if (!check(f(sm, seg[l]))) {
        while (l < size) {
          l = (2 * l);
          if (check(f(sm, seg[l]))) {
            sm = f(sm, seg[l]);
            l++;
          }
        }
        return l - size;
      }
      sm = f(sm, seg[l]);
      l++;
    } while ((l & -l) != l);
    return N;
  }

  // check(a[l] * ... * a[r-1]) が true となる最小の l
  // (左端まで true なら 0 を返す)
  template <typename C>
  int min_left(int r, C check) {
    assert(0 <= r && r <= N);
    assert(check(I) == true);
    if (r == 0) return 0;
    r += size;
    T sm = I;
    do {
      r--;
      while (r > 1 && (r % 2)) r >>= 1;
      if (!check(f(seg[r], sm))) {
        while (r < size) {
          r = (2 * r + 1);
          if (check(f(seg[r], sm))) {
            sm = f(seg[r], sm);
            r--;
          }
        }
        return r + 1 - size;
      }
      sm = f(seg[r], sm);
    } while ((r & -r) != r);
    return 0;
  }
};





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 G>
struct Tree {
 private:
  G& g;
  int root;
  vector<array<int, 24>> bl;
  vector<int> dp;
  void build() {
    bl.resize(g.size());
    dp.resize(g.size());
    for (auto& v : bl) fill(begin(v), end(v), -1);
    dfs(root, -1, 0);
  }

  void dfs(int c, int p, int _dp) {
    dp[c] = _dp;
    for (int i = p, x = 0; i != -1;) {
      bl[c][x] = i;
      i = bl[i][x], x++;
    }
    for (auto& d : g[c]) {
      if (d == p) continue;
      dfs(d, c, _dp + 1);
    }
  }

 public:
  Tree(G& _g, int _r = 0) : g(_g), root(_r) { build(); }

  int depth(int u) const { return dp[u]; }

  int par(int u) const { return u == root ? -1 : bl[u][0]; }

  int kth_ancestor(int u, int k) const {
    if (dp[u] < k) return -1;
    while (k) {
      int t = __builtin_ctz(k);
      u = bl[u][t], k ^= 1 << t;
    }
    return u;
  }

  int nxt(int s, int t) const {
    if (dp[s] >= dp[t]) return par(s);
    int u = kth_ancestor(t, dp[t] - dp[s] - 1);
    return bl[u][0] == s ? u : bl[s][0];
  }

  vector<int> path(int s, int t) const {
    vector<int> pre, suf;
    while (dp[s] > dp[t]) {
      pre.push_back(s);
      s = bl[s][0];
    }
    while (dp[s] < dp[t]) {
      suf.push_back(t);
      t = bl[t][0];
    }
    while (s != t) {
      pre.push_back(s);
      suf.push_back(t);
      s = bl[s][0];
      t = bl[t][0];
    }
    pre.push_back(s);
    reverse(begin(suf), end(suf));
    copy(begin(suf), end(suf), back_inserter(pre));
    return pre;
  }

  int lca(int u, int v) {
    if (dp[u] != dp[v]) {
      if (dp[u] > dp[v]) swap(u, v);
      v = kth_ancestor(v, dp[v] - dp[u]);
    }
    if (u == v) return u;
    for (int i = __lg(dp[u]); i >= 0; --i) {
      if (dp[u] < (1 << i)) continue;
      if (bl[u][i] != bl[v][i]) u = bl[u][i], v = bl[v][i];
    }
    return bl[u][0];
  }

  // u - v 間のパス上の頂点のうち u から距離 i の頂点
  // (dist(u, v) < i のとき -1)
  int jump(int u, int v, int i) {
    int lc = lca(u, v);
    int d1 = dp[u] - dp[lc];
    if (i <= d1) return kth_ancestor(u, i);
    int d = d1 + dp[v] - dp[lc];
    if (i <= d) return kth_ancestor(v, d - i);
    return -1;
  }
};

/**
 * @brief 木に対する一般的なクエリ
 * @docs docs/tree/tree-query.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(int n, int r) {
    if (n < 0 || r < 0) return T(0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};


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

using namespace Nyaan;

void q() {
  inl(N, M);
  auto G = graph(N, M);

  vvi g(N);
  {
    vi mx = mkiota(N);
    UnionFind uf(N);
    rep(i, N) each(j, G[i]) {
      if (j < i and !uf.same(i, j)) {
        int k = mx[uf.find(j)];
        g[i].push_back(k);
        g[k].push_back(i);
        uf.unite(i, k);
        mx[uf.find(i)] = i;
      }
    }
  }
  // trc(g);

  HeavyLightDecomposition hld(g, N - 1);
  Tree tree(g, N - 1);
  SegmentTree dp(
      N, [](mint a, mint b) { return a + b; }, mint{});

  rep(i, N) {
    mint cur = 1;

    vi chds;
    each(j, G[i]) {
      if (j < i) chds.push_back(j);
    }
    sort(all(chds), [&](int s, int t) { return hld.down[s] < hld.down[t]; });
    trc(chds);

    // aux tree
    vi rs;
    rs.push_back(i);
    each(c, chds) {
      int l = hld.lca(rs.back(), c);
      trc(c, l, rs);
      while (hld.down[l] < hld.down[rs.back()] and
             hld.down[rs.back()] < hld.up[l]) {
        rs.pop_back();
        assert(!rs.empty());
      }
      rs.push_back(c);
      assert(l != c);
      int nxt = tree.nxt(l, c);
      hld.path_query(nxt, c, true,
                     [&](int u, int v) { cur += dp.query(u, v); });
    }
    dp.update(hld.idx(i).fi, cur);
  }

  // rep(i, N) cerr << dp[i] << " \n"[i + 1 == N];
  out(dp.query(0, N));
}

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