Problem p02822 (C++)

Buggy submission (status: Runtime Error):
#include <algorithm>
#include <bits/stdc++.h>
#include <cassert>
#include <complex>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <iostream>
#include <list>
#include <map>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <stdio.h>
#include <string>
#include <vector>
using namespace std;
typedef long long ll;
typedef long double ld;
using edge = struct {
  ll to;
  ll cost;
};
struct point {
  ll x;
  ll y;
  bool operator<(const point &p) const {
    if (x == p.x)
      return y < p.y;
    return x < p.x;
  }
};
struct undirected_edge {
  ll from;
  ll to;
  ll cost;
  bool operator<(const undirected_edge &ue) const { return cost < ue.cost; }
};

typedef string str;
typedef std::pair<ll, ll> pl;
typedef std::pair<ll, pl> pl3;
typedef std::map<string, ll> msl;
typedef std::map<char, ll> mcl;
typedef std::map<ll, ll> mll;
typedef std::vector<ll> vl;
typedef std::vector<pl> vpl;
typedef std::vector<point> points;
typedef std::vector<pl3> vpl3;
typedef std::priority_queue<ll> pq;
typedef std::priority_queue<ll, vl, greater<ll>>
    pql; // priority queue taking from the lower value.
typedef std::vector<edge> Graph;
const ll N_DIGITS = 60;
const ll MOD = 1e9 + 7;
// const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-9;
const long double PI = 3.14159265358979323846;
vpl dirs = {
    {-1, 0}, {1, 0},  {0, 1},   {0, -1}, // four directions
    {1, 1},  {-1, 1}, {-1, -1}, {1, -1}, // diagonal
    {0, 0}                               // self
};

template <typename A, typename B> string to_string(pair<A, B> p);

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);

string to_string(const string &s) { return '"' + s + '"'; }

string to_string(const char *s) { return to_string((string)s); }

string to_string(bool b) { return (b ? "true" : "false"); }

string to_string(vector<bool> v) {
  bool first = true;
  string res = "{";
  for (int i = 0; i < static_cast<int>(v.size()); i++) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(v[i]);
  }
  res += "}";
  return res;
}

template <size_t N> string to_string(bitset<N> v) {
  string res = "";
  for (size_t i = 0; i < N; i++) {
    res += static_cast<char>('0' + v[i]);
  }
  return res;
}

template <typename A> string to_string(A v) {
  bool first = true;
  string res = "{";
  for (const auto &x : v) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(x);
  }
  res += "}";
  return res;
}

template <typename A, typename B> string to_string(pair<A, B> p) {
  return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " +
         to_string(get<2>(p)) + ")";
}

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " +
         to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}

void debug_out() { cerr << endl; }

template <typename Head, typename... Tail> void debug_out(Head H, Tail... T) {
  cerr << " " << to_string(H);
  debug_out(T...);
}

#define rep(i, n) for (ll(i) = 0; (i) < (n); (i)++)
#define revrep(i, n) for (ll(i) = n - 1; (i) >= 0; (i)--)
#define For(i, a, b) for (ll(i) = (a); (i) < (b); (i)++)
#define revFor(i, b, a) for (ll(i) = (b)-1; (i) >= (a); (i)--)
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define isUpper(c) ('a' - c > 0)
#define isNum(c) (0 <= (c) - '0' && (c) - '0' <= 9)
#define toLower(c) char((c) + 0x20)
#define toUpper(c) char((c)-0x20)
#define pb push_back
#define mp make_pair
#define pr(a) cout << (a)
#define prl(a) cout << (a) << endl
#define prl2(a, b) cout << (a) << " " << (b) << endl
#define prl3(a, b, c) cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a, b, c, d)                                                       \
  cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) cout << (a) << " "
#define prs2(a, b) cout << (a) << " " << (b) << " "
#define prs3(a, b, c) cout << (a) << " " << (b) << " " << (c) << " "
#define prs4(a, b, c, d)                                                       \
  cout << (a) << " " << (b) << " " << (c) << " " << (d) << " "
#define yn(condition)                                                          \
  if ((condition))                                                             \
    prl("Yes");                                                                \
  else                                                                         \
    prl("No");
#define YN(condition)                                                          \
  if ((condition))                                                             \
    prl("YES");                                                                \
  else                                                                         \
    prl("NO");
#define in1(a) cin >> (a)
#define in2(a, b) cin >> (a) >> (b)
#define in3(a, b, c) cin >> (a) >> (b) >> (c)
#define in4(a, b, c, d) cin >> (a) >> (b) >> (c) >> (d)
#define e1 first
#define e2 second
#define ctol(c) ll((c)) - ll('0')
#define ltos(n) to_string((n))
#define items(kv, v) for (auto &(kv) : (v))
#define ndig(N, n) ctol(ll(ltos((N))[ll(ltos((N)).length()) - (n)]))
#define rsort(a, n) sort(a, a + n, greater<>())
#define Forchar(c, a, z) for (char(c) = (a); (c) <= (z); (c)++)
#define cntchar(s, c) count(all((s)), c)
#define substring(s, start, end) s.substr((start), (end) - (start) + 1)
#define prl_nd(num, digits)                                                    \
  cout << fixed << setprecision(digits) << (num) << endl;
#define XOR(a, b) (a) ^ (b)
#define prl_time(s)                                                            \
  prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))
#define DEBUG(x) cerr << #x << ": " << (x) << endl
#define DEBUG_VEC(v)                                                           \
  cerr << #v << ": ";                                                          \
  rep(__i, (v).size()) cerr << ((v)[__i]) << ", ";                             \
  cerr << endl
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)

template <class T> inline bool chmax(T &a, T b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}

template <class T> inline bool chmin(T &a, T b) {
  if (a > b) {
    a = b;
    return true;
  }
  return false;
}

struct MaxFlow {
  struct F_edge {
    ll to, rev, capacity;
    F_edge(ll to, ll rev, ll capacity) : to(to), rev(rev), capacity(capacity) {}
  };

  typedef vector<F_edge> F_edges;
  vector<F_edges> graph;
  ll n_vertex;
  // level is the shortest path to get a given node from the source node.
  vl level, iter;

  MaxFlow(ll n_vertex) : n_vertex(n_vertex) { graph.resize(n_vertex); }

  void add_edge(ll from, ll to, ll capacity) {
    graph[from].pb({to, ll(graph[to].size()), capacity});
    graph[to].pb({from, ll(graph[from].size()) - 1, 0});
  }

  void bfs(ll source) {
    level = vl(n_vertex, -1);
    level[source] = 0;
    queue<ll> q;
    q.push(source);
    while (!q.empty()) {
      ll vertex = q.front();
      q.pop();
      rep(i, graph[vertex].size()) {
        ll target = graph[vertex][i].to;
        ll cap_target = graph[vertex][i].capacity;
        // if the flow can be into the target node, implement below.
        if (cap_target > 0 && level[target] < 0) {
          level[target] = level[vertex] + 1;
          q.push(target);
        }
      }
    }
  }

  ll dfs(ll vertex, ll sink, ll flow) {
    if (vertex == sink)
      return flow;
    for (ll &i = iter[vertex]; i < graph[vertex].size(); i++) {
      ll target = graph[vertex][i].to;
      ll cap_target = graph[vertex][i].capacity;
      ll rev_target = graph[vertex][i].rev;
      // if capasitiy is not full yet and target is farther,
      // then assign current flow
      if (cap_target > 0 && level[vertex] < level[target]) {
        ll d = dfs(target, sink, min(cap_target, flow));
        if (d > 0) { // if the flow successfully reaches the sink, reduce the
                     // flow from the capacity
          graph[vertex][i].capacity -= d;
          graph[target][rev_target].capacity += d;
          return d;
        }
      }
    }
    return 0;
  }

  ll dinic(ll source, ll sink) {
    ll flow = 0;
    while (true) {
      bfs(source);
      // if there is no path leading to the sink, the maximum flow is 0.
      if (level[sink] < 0)
        return flow;
      iter = vl(n_vertex, 0);
      ll f;
      while ((f = dfs(source, sink, INF)) > 0)
        flow += f;
    }
  }
};

class UnionFind {
  vl parents, set_size;

public:
  UnionFind() {}
  UnionFind(ll n) {
    parents = set_size = vl(n);
    rep(i, n) {
      parents[i] = i;
      set_size[i] = 1LL;
    }
  }

  ll root_find(ll x) {
    if (parents[x] == x)
      return x;
    return parents[x] = root_find(parents[x]);
  }

  void unite(ll x, ll y) {
    // priority for x is larger than that of y
    x = root_find(x);
    y = root_find(y);
    if (x == y)
      return;
    parents[y] = x, set_size[x] += set_size[y];
  }

  bool is_same(ll x, ll y) { // connected or not
    return root_find(x) == root_find(y);
  }

  ll size(ll x) { return set_size[root_find(x)]; }
};

struct Doubling { // ABC167D
  ll n;
  ll sz;
  vector<vl> next;
  /*
      next[k + 1][i] := next[k][next[k][i]]
      next[0][i] := edge[i]
      e.g. a0, a1, ..., an-1 / 0 <= ai <= n - 1
      a0 -> a[a0] -> a[a[a0]] -> ... -> a[a[...[a[0]]]] (m times)
      Let the function repeatedly input a[i] m times be f[m](a[i])

      - get(i, x) returns f[x](a[i])
      - lower_bound(i, j) returns minimum x which satisfies f[x](a[i]) >= j.
                                  if not possible returns n.

  */

  // edge[i]: the step size for one iteration
  Doubling(vl &edge) : n(edge.size()), sz(62) {
    next.resize(sz, vl(n, -1));
    rep(i, n) next[0][i] = edge[i];
    rep(k, sz - 1) rep(i, n) next[k + 1][i] = next[k][next[k][i]];
  }

  ll get(ll i, ll x) {
    ll ret = i;
    rep(bit, sz) {
      if (!(x >> bit & 1))
        continue;
      ret = next[bit][ret];
    }
    return ret;
  }

  ll lower_bound(ll i, ll j) {
    ll cur = i, acc = 0;
    revrep(wid, sz) {
      if (next[wid][cur] < j) {
        acc += 1LL << wid;
        cur = next[wid][cur];
      }
    }
    return min(n, acc + 1);
  }
};

/*
class LCA{
public:
    ll N, logN;
    vl depth, len;
    Graph tree[200005]; // global declaration later.
    vector<vl> parents;

    LCA(ll n){
        N = n;
        logN = 0;
        while (N > (1LL << logN)) logN++;
        depth = vl(N); len = vl(N);
        parents = vector<vl>(logN, vl(N));
        init(0, -1, 0, 0);
        build();
    }

    void init(ll source, ll parent, ll d, ll l){
        depth[source] = d;
        parents[0][source] = parent;
        len[source] = l;
        rep(i, tree[source].size()){
            ll target = tree[source][i].to;
            ll cost = tree[source][i].cost;
            if (target == parent) continue;
            init(target, source, d + 1, cost + l);
        }
    }

    void build(){
        rep(k, logN - 1) rep(n, N){
            // if there is no parent, -1.
            // otherwise, the parent of the parent is the parent.
            if (parents[k][n] < 0) parents[k + 1][n] = -1;
            else parents[k + 1][n] = parents[k][parents[k][n]];
        }
    }

    ll query(ll u, ll v){
        if (depth[u] > depth[v]) swap(u, v);
        rep(k, logN) if ((depth[v] - depth[u]) >> k & 1) v = parents[k][v];
        if (u == v) return u;

        revrep(k, logN){
            if (parents[k][u] != parents[k][v]){
                u = parents[k][u]; v = parents[k][v];
            }
        }
        return parents[0][u];
    }

    ll distance(ll u, ll v){
        ll w = query(u, v);
        return len[u] + len[v] - 2 * len[w];
    }

};
*/

struct BIT {
  ll n;
  vl dat;
  BIT(ll n, ll ini = 0) : dat(n + 1, ini), n(n){};

  //       x: 1001 1010  1100 1011 1101  1111
  // x & - x: 0001 0010  0100 0001 0001  0001
  //      ->: 1010 1100 10000 1100 1100 10000

  ll update_func(ll val, ll d) {
    // if maximum -> max(val, dat)
    // return max(val, dat);
    // if cumulative sum
    return val + d;
  }

  ll query(ll i) {
    /*
    v[0] + v[1] + ... + v[i]
    e.g.) i = 10101
    itr1. 10101 -> 10100
    itr2. 10100 -> 10000
    itr3. 10000 -> 00000 (break)
    */
    ll ret = 0;
    for (ll j = i; j >= 0; j = (j & (j + 1)) - 1) {
      ret = update_func(ret, dat[j]);
    }
    return ret;
  }

  ll query(ll l, ll r) { return query(r) - query(l); }

  ll lower_bound(ll key) {
    // v[0] + v[1] + ... + v[left - 1] < key <= v[0] + v[1] + ... + v[left]
    if (key <= 0)
      return 0;
    ll left = 0, right = 1;
    while (right <= n)
      right *= 2;

    for (ll i = right; i > 0; i /= 2) {
      if (left + i <= n && dat[left + i - 1] < key) {
        key -= dat[left + i - 1];
        left += i;
      }
    }
    return left;
  }

  void update(ll i, ll val) {
    /*
    e.g.) i = 10101, n = 11111
    itr1. i: 10101, i+1: 10110 -> 10111
    itr2. i: 10111, i+1: 11000 -> 11111 (break)
    */
    if (i < 0)
      return;
    for (ll j = i; j < n; j |= j + 1) {
      dat[j] = update_func(val, dat[j]);
    }
  }
};

struct SegmentTree {
  ll n, ini, minimize;
  vl dat;

  // when seeking minimum
  // ini = INF
  // when seeking maximum
  // ini = -INF
  SegmentTree(ll n_, bool minimize_ = true) {
    n = 1;
    minimize = minimize_;
    if (minimize)
      ini = INF;
    else
      ini = -INF;
    while (n < n_)
      n *= 2;
    dat.resize(2 * n - 1);
    rep(i, 2 * n - 1) dat[i] = ini;
  };

  void update(ll idx, ll val) {
    idx += n - 1;
    if (minimize && dat[idx] <= val)
      return;
    if (!minimize && dat[idx] >= val)
      return;

    dat[idx] = val;
    while (idx > 0) {
      idx = (idx - 1) / 2;
      // when seeking minimum
      if (minimize)
        dat[idx] = min(dat[idx * 2 + 1], dat[idx * 2 + 2]);
      // when seeking maximum
      else
        dat[idx] = max(dat[idx * 2 + 1], dat[idx * 2 + 2]);
    }
  }

  ll query(ll l, ll r) {
    // ### NOTE ###
    // the range is [l, r]
    // l, l + 1, ..., r
    r++; // to adjust to this method
    return query_segment(l, r, 0, 0, n);
  }

  ll query_segment(ll a, ll b, ll idx, ll l, ll r) {
    assert(a < b);

    if (r <= a || b <= l)
      return ini;
    if (a <= l && r <= b)
      return dat[idx];
    else {
      ll seg1 = query_segment(a, b, idx * 2 + 1, l, (l + r) / 2);
      ll seg2 = query_segment(a, b, idx * 2 + 2, (l + r) / 2, r);
      // when seeking minimum
      if (minimize)
        return min(seg1, seg2);
      // when seeking maximum
      else
        return max(seg1, seg2);
    }
  }
};

template <class Target> class RerootingTreeDP {
public:
  using T = typename Target::type;
  struct DP_edge {
    ll to, rev; // rev is the index to trace the source node.
    T value;    // objective value
  };

private:
  ll n;
  void dfs_fwd(ll source, ll parent) {
    ll par_idx = -1;
    vector<T> values;
    rep(i, tree[source].size()) {
      const DP_edge &e = tree[source][i];
      if (e.to == parent) {
        par_idx = i;
        continue;
      }
      dfs_fwd(e.to, source);
      values.pb(e.value);
    }

    // If the parent != -1, update the value on edge from parent to source
    if (par_idx != -1) {
      ll src_idx = tree[source][par_idx].rev;
      // update values on the edge from parent to source
      tree[parent][src_idx].value = Target::merge(values);
    }
  }

  void dfs_bwd(ll source, ll parent) {
    vector<T> values;
    for (auto &&e : tree[source])
      values.pb(e.value);
    values = Target::evaluate(values);

    rep(i, tree[source].size()) {
      const DP_edge &e = tree[source][i];
      if (e.to == parent)
        continue;
      // tree[e.to][e.rev]: e.to -> source
      tree[e.to][e.rev].value = values[i];
      dfs_bwd(e.to, source);
    }
  }

public:
  UnionFind uf;
  vector<vector<DP_edge>> tree;
  RerootingTreeDP(ll n) : n(n), uf(n), tree(n) {}

  void add_edge(ll u, ll v, T val) {
    assert(!uf.is_same(u, v));
    tree[u].pb({v, ll(tree[v].size()), val});
    tree[v].pb({u, ll(tree[u].size()) - 1, val});
    uf.unite(u, v);
  }

  void dp() {
    vector<bool> visited(n, false);
    rep(i, n) {
      if (visited[uf.root_find(i)])
        continue;
      dfs_fwd(i, -1);
      visited[uf.root_find(i)] = true;
    }

    visited.assign(n, false);
    rep(i, n) {
      if (visited[uf.root_find(i)])
        continue;
      dfs_bwd(i, -1);
      visited[uf.root_find(i)] = true;
    }
  }

  ll size() const { return tree.size(); }
};

struct StronglyConnectedComponents {
  ll n;
  vector<vl> graph, graph_rev, dag, cmp;
  vl order, visited, cmp_idx;

  StronglyConnectedComponents() {}
  StronglyConnectedComponents(ll sz)
      : n(sz), graph(sz), graph_rev(sz), visited(sz), cmp_idx(sz) {}

  void add_edge(ll from, ll to) {
    graph[from].pb(to);
    graph_rev[to].pb(from);
  }

  void input(ll m, ll offset = -1) {
    ll a, b;
    rep(i, m) {
      in2(a, b);
      add_edge(a + offset, b + offset);
    }
  }

  ll operator[](ll k) { return cmp_idx[k]; }

  void dfs_fwd(ll source) {
    visited[source] = 1;
    rep(i, graph[source].size()) {
      ll target = graph[source][i];
      if (!visited[target])
        dfs_fwd(target);
    }
    order.pb(source);
  }

  void dfs_bwd(ll source, ll num) {
    visited[source] = 1, cmp_idx[source] = num;

    cmp[num].pb(source);
    rep(i, graph_rev[source].size()) {
      ll target = graph_rev[source][i];
      if (!visited[target])
        dfs_bwd(target, num);
    }
  }

  ll build() {
    fill(all(visited), 0);
    order.clear();
    rep(i, n) if (!visited[i]) dfs_fwd(i);

    fill(all(visited), 0);
    ll num = 0;
    revrep(i, order.size()) {
      if (!visited[order[i]]) {
        dag.pb(vl());
        cmp.pb(vl());
        dfs_bwd(order[i], num++);
      }
    }

    rep(i, n) for (ll to : graph[i]) if (cmp_idx[i] != cmp_idx[to])
        dag[cmp_idx[i]]
            .pb(cmp_idx[to]);

    rep(i, num) {
      sort(all(dag[i]));
      dag[i].erase(unique(all(dag[i])), dag[i].end());
    }
    return num;
  }
};

ll gcd(ll m, ll n) {
  ll a = max(m, n);
  ll b = min(m, n);
  while (b != 1 && b != 0) {
    a %= b;
    swap(a, b);
  }
  return b == 1 ? 1 : a;
}

ll lcm(ll m, ll n) { return m / gcd(m, n) * n; }

ll power_mod(ll a, ll power, ll mod) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = (value * a) % mod;
    a = (a * a) % mod;
    power = power >> 1;
  }
  return value % mod;
}

ll modinv(ll a, ll mod) { return power_mod(a, mod - 2, mod); }

ll power_normal(ll a, ll power) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = value * a;
    a = a * a;
    power = power >> 1;
  }
  return value;
}

ll comb_memo[55][55];

ll pascal_triangle(ll n) {
  comb_memo[0][0] = 1;
  For(i, 1, n + 1) rep(j, i + 1) {
    comb_memo[i][j] += comb_memo[i - 1][j];
    if (j > 0)
      comb_memo[i][j] += comb_memo[i - 1][j - 1];
  }
}

ll combination(ll n, ll r, ll mod) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    ll num = (n - i) % mod, den = (i + 1) % mod;
    (numerator *= num) %= mod;
    (denomenator *= modinv(den, mod)) %= mod;
  }

  return (numerator * denomenator) % mod;
}

ll combination_no_mod(ll n, ll r) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    numerator *= n - i;
    denomenator *= i + 1;
    ll g = gcd(numerator, denomenator);
    numerator /= g;
    denomenator /= g;
  }

  return numerator;
}

ld log_combination(ll n, ll r) {
  if (n == r && n == 0)
    return 0;
  else if (n <= 0 || r < 0 || r > n)
    return -INF;
  ld val = 0;
  for (ll i = 0; i < r; i++) {
    val += log(n - i);
    val -= log(i + 1);
  }

  return val;
}

ll bin_search(ll key, ll A[], ll left, ll right) {
  // return the index idx where A[idx] = key.
  // A[left] is start and A[right] is end..
  // In other words, A[right], not A[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (A[mid] == key)
      return mid;
    else if (A[mid] > key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid + 1;
  }
  return -1;
}

/*
ll bin_search_temp(ll left, ll right, callable judge){
    while(right > left){
        // when seeking lower bound
        ll mid = (right + left) / 2;
        if (judge(mid)) right = mid;
        else left = mid + 1;
        // when seeking upper bound
        ll mid = (right + left + 1) / 2;
        if (judge(mid)) left = mid;
        else right = mid - 1;
    }

    return right;
}

trinary_search

// Care the value EPS!!! Compare to the condition
ld left = 0; ld right = p;
while(abs(right - left) > EPS){
    ld left2 = (2 * left + right) / 3.0;
    ld right2 = (left + 2 * right) / 3.0;
    ld f1 = tak_func(left2);
    ld f2 = tak_func(right2);
    if (f1 <= f2) right = right2;
    else if (f2 <= f1) left = left2;
}

*/

str bin_expression(ll n, ll dig) {
  str s = "";
  rep(i, dig) {
    s += ltos(n % 2);
    n /= 2;
  }
  reverse(all(s));
  return s;
}

ll lower_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] >= key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] < key && key <= A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] < key && key <= A[mid])
      return mid;
    else if (A[mid - 1] >= key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid;
  }

  return -1; // all the elements < key
}

ll inf_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] > key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] <= key && key < A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] <= key && key < A[mid])
      return mid;
    else if (A[mid - 1] > key)
      right = mid - 1;
    else if (A[mid] <= key)
      left = mid;
  }

  return -1; // all the elements <= key
}

ll upper_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] <= key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] <= key && key < A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] <= key && key < A[mid + 1])
      return mid;
    else if (A[mid] > key)
      right = mid;
    else if (A[mid + 1] <= key)
      left = mid + 1;
  }

  return -1; // all the elements > key
}

ll sup_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] < key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] < key && key <= A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] < key && key <= A[mid + 1])
      return mid;
    else if (A[mid] >= key)
      right = mid;
    else if (A[mid + 1] < key)
      left = mid + 1;
  }

  return -1; // all the elements >= key
}

ll bin_search_vector(ll key, vl v, ll left, ll right) {
  // return the index idx where v[idx] = key.
  // v[left] is start and v[right] is end..
  // In other words, v[right], not v[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (v[mid] == key)
      return mid;
    else if (v[mid] > key)
      right = mid - 1;
    else if (v[mid] < key)
      left = mid + 1;
  }
  return -1;
}

ll lower_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] < key <= v[N]
  // N == -1 if all the elements < key
  return lower_bound(all(v), key) - v.begin();
}

ll inf_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] <= key < v[N] <= key + 1
  // N == -1 if all the elements <= key
  return lower_bound(all(v), key + 1) - v.begin();
}

ll upper_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key < v[N + 1]
  // N == -1 if all the elements > key
  return upper_bound(all(v), key) - v.begin(); // (- 1)
}

ll sup_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key - 1 < key <= v[N + 1]
  // N == -1 if all the elements >= key
  return upper_bound(all(v), key - 1) - v.begin() - 1;
}

ll fact(ll n) {
  if (n == 0)
    return 1;
  return n * fact(n - 1);
}

ll fact_mod(ll n, ll mod) {
  if (n == 0)
    return 1;
  return n * fact_mod(n - 1, mod);
}

/*
ll facts[6000], facts_inv[6000], inv[6000];

void fact_memo(ll n, ll mod){
    facts[0] = facts[1] = 1;
    facts_inv[0] = facts_inv[1] = 1;
    inv[1] = 1;

    For(i, 2, n + 1){
        facts[i] = (i * facts[i - 1]) % mod;
        inv[i] = mod - inv[mod % i] * (mod / i) % mod;
        facts_inv[i] = facts_inv[i - 1] * inv[i] % mod;
    }
}

ll nCk(ll n, ll r){
    if (n == r && n == 0) return 1;
    else if (n <= 0 || r < 0 || r > n) return 0;
    ll val = (facts[n] * facts_inv[n - r]) % MOD;
    val *= facts_inv[r];
    return val % MOD;
}
*/

bool is_prime(ll n) {
  if (n <= 1)
    return false;
  for (ll i = 2; i * i <= n; i++) {
    if (n % i == 0)
      return false;
  }
  return true;
}

ll bool_sum(ll a1, ll a2) {
  if (a1 == 1 || a2 == 1)
    return 1;
  return 0;
}

mll prime_factorization(ll n) {
  ll i = 2;
  mll table;
  while (i * i <= n) {
    while (n % i == 0) {
      table[i]++;
      n /= i;
    }
    i++;
  }
  if (n > 1)
    table[n] = 1;
  return table;
}

vl divisor_table(ll n) {
  vl table;
  ll i = 1;
  while (i * i <= n) {
    if (n % i == 0) {
      table.pb(i);
      if (i * i != n)
        table.pb(n / i);
    }
    i++;
  }

  sort(all(table));

  return table;
}

ll char_to_idx(char c) {
  ll idx = 0;
  Forchar(cc, 'a', 'z') {
    if (c == cc)
      return idx;
    else
      idx++;
  }
}

ll next_combination(ll sub) {
  /*
  ### Attention ###
  if k is 0 or 1 and n is 0, it does not work well.

  ll n, k; ll bit = (1 << k) - 1;
  for (; bit < (1 << n); bit = next_combination(bit)){
      bool ith = bit & (1 << i);
      procedures...
  }

      sub & -sub: the binary which shares the last digit whose value is 1 in sub
      sub + x   : carry up the last digit
          ~y    : the binary whose digits are 1 if y's digit is 0.
  (sub & ~y) / x: reduce the same number of 0s after first 1 in x from (sub &
  ~y).
  */
  ll x = sub & -sub, y = sub + x;
  return (((sub & ~y) / x) >> 1) | y;
}

// just change the input if you want to change the target.
// If you want to check the common sequences in two strings,
// combine them. e.g. ABC150F
vl z_algorithm(str s) {
  /*
  Paramters
  ---------
  str: the string of interest

  Returns
  -------
  res[i] is the maximum number of K which satisfies
  s[:K] == s[i:i + K]
  for each i = 0, 1, 2, ..., n - 1.
  */
  ll n = s.length();
  vl res(n);
  res[0] = n;
  ll i1 = 1, i2 = 0;

  while (i1 < n) {
    /*
    i1: the starting point
    i2: the length of substring
    */
    while (i1 + i2 < n && s[i2] == s[i1 + i2])
      ++i2;
    res[i1] = i2;
    if (i2 == 0) {
      ++i1;
      continue;
    }
    ll i3 = 1;

    // update the already seen points
    while (i1 + i3 < n && i3 + res[i3] < i2) {
      res[i1 + i3] = res[i3];
      ++i3;
    }
    // update up to i1 + i3 and the next possible minimum length is i2 - i3 (=
    // res[i3])
    i1 += i3, i2 -= i3;
  }
  return res;
}

ll big_number_mod(str s, ll mod) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth) % mod;
    val %= mod;
    tenth *= 10;
    tenth %= mod;
    idx++;
  }

  return val;
}

ll big_number_compare(str s1, str s2) {
  if (s1.length() > s2.length())
    return 1;
  else if (s1.length() < s2.length())
    return -1;
  else if (s1 == s2)
    return 0;

  return 2 * (s1 > s2) - 1;
}

ll string_to_ll(str s) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth);
    tenth *= 10;
    idx++;
  }

  return val;
}

str reflected_string(str s) {
  str t, u;
  ll n = s.length();
  t = s;
  reverse(all(t));
  u = substring(t, 0, n - 2) + s + substring(t, 1, n - 1);
  return u;
}

ld distance_between_point_line(point l_begin, point l_end, point p) {
  ll xl1 = l_begin.x, yl1 = l_begin.y;
  ll xl2 = l_end.x, yl2 = l_end.y;
  ll xp = p.x, yp = p.y;
  ll a = yl2 - yl1;
  ll b = -xl2 + xl1;
  ll c = -a * xl2 - b * yl2;
  return abs(ld(a * xp + b * yp + c)) / ld(sqrt(a * a + b * b));
}

bool is_cross(point l1_begin, point l1_end, point l2_begin, point l2_end) {
  ll x1 = l1_begin.x, y1 = l1_begin.y;
  ll x2 = l1_end.x, y2 = l1_end.y;
  ll x3 = l2_begin.x, y3 = l2_begin.y;
  ll x4 = l2_end.x, y4 = l2_end.y;
  ll val1 = (x1 - x2) * (y3 - y1) + (y1 - y2) * (x1 - x3);
  ll val2 = (x1 - x2) * (y4 - y1) + (y1 - y2) * (x1 - x4);
  ll val3 = (x3 - x4) * (y1 - y3) + (y3 - y4) * (x3 - x1);
  ll val4 = (x3 - x4) * (y2 - y3) + (y3 - y4) * (x3 - x2);

  return val1 * val2 < 0 && val3 * val4 < 0;
}

ld space_of_triangle(point p1, point p2, point p3) {
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;
  ll v1 = x2 - x1;
  ll u1 = y2 - y1;
  ll v2 = x3 - x1;
  ll u2 = y3 - y1;
  ld s = ld(v1 * u2 - u1 * v2) / ld(2);
  return abs(s);
}

pair<point, ll> center_2det_of_3points(point p1, point p2, point p3) {
  // the center of circle on the given three points
  // return the determinant value and the product of center points and 2 *
  // determinant value
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;

  // center of 2 points
  ll c2x_2 = x2 + x1, c2y_2 = y2 + y1;
  ll c3x_2 = x3 + x1, c3y_2 = y3 + y1;

  // vertical vector of 2 lines
  ll b2y = x2 - x1, b2x = -y2 + y1;
  ll b3y = x3 - x1, b3x = -y3 + y1;

  ll x1_2 = c3x_2 * b3y - c3y_2 * b3x, y1_2 = c2x_2 * b2y - c2y_2 * b2x;
  ll det = -b3y * b2x + b3x * b2y;
  if (det == 0)
    return {{INF, INF}, det};
  ll cx_2det = -b2x * x1_2 + b3x * y1_2, cy_2det = -b2y * x1_2 + b3y * y1_2;
  return {{cx_2det, cy_2det}, det};
}

ll inversion_number(vl a, ll a_max) {
  /*
  Paramters
  ---------
  a: vector<ll>
      All the elements must be non-negative.
      Prefably the elements are compressed to reduce the computational cost.
  a_max: ll
      The maximum value of the vector a or the value bigger than the value
  stated previously.
  */

  BIT bit(a_max + 1);
  ll val = 0;
  rep(i, a.size()) {
    // i is the number of elements that have lower index than a[i].
    // call the number of elements that have lower value than a[i]
    // by subtracting these two, the residual number is the number of elements
    // that have larger value
    val += i - bit.query(a[i] - 1); // cumulative sum from 0 to a[i] - 1
    bit.update(a[i], 1);
  }
  return val;
}

template <typename T> vector<T> compress(vector<T> v) {
  // sort and remove all the duplicated values
  sort(all(v));
  v.erase(unique(all(v)), v.end());
  return v;
}

template <typename T> map<T, ll> dict(const vector<T> &v) {
  map<T, ll> d;
  rep(i, v.size()) d[v[i]] = i;
  return d;
}

points compress2D(vl xs, vl ys) {
  /*
  NOTE
  ----
  Add the corner points if required
  */

  ll n = xs.size();
  vl xcs = compress(xs), ycs = compress(ys);
  map<ll, ll> xd = dict(xcs), yd = dict(ycs);
  points ps(n);

  rep(i, n) xs[i] = xd[xs[i]], ys[i] = yd[ys[i]];

  rep(i, n) ps[i] = {xs[i], ys[i]};

  sort(all(ps));

  return ps;
}

void GaussJordanBitVector(vl &bs) {
  ll n = bs.size();
  ll rank = 0;
  ll j = 0;
  revrep(i, N_DIGITS) {
    for (j = rank; j < n; j++)
      if (bs[j] & (1LL << i))
        break;
    if (j == n)
      continue;
    if (j > rank)
      bs[rank] ^= bs[j];
    for (j = rank + 1; j < n; j++)
      bs[j] = min(bs[j], bs[j] ^ bs[rank]);
    rank++;
  }
}

ll kruskal(vector<undirected_edge> &es, ll n_vertex) {
  sort(all(es));
  UnionFind uf(n_vertex);

  ll min_cost = 0;

  rep(i, es.size()) {
    undirected_edge &e = es[i];
    if (!uf.is_same(e.from, e.to)) {
      min_cost += e.cost;
      uf.unite(e.from, e.to);
    }
  }
  return min_cost;
}

ll LongestIncreasedSequence(vl &v, ll n) {
  vl dp(n, INF);
  rep(i, n) * lower_bound(all(dp), v[i]) = v[i];
  return find(all(dp), INF) - dp.begin();
}

/*

diameter of tree
Graph tree[MAX_N];
ll dM = 0, vM = 0, v2 = 0;

void dfs1(ll source, ll parent, ll d){
    if (d > dM) dM = d, vM = source;

    rep(i, tree[source].size()){
        ll target = tree[source][i].to;
        if (target == parent) continue;
        dfs1(target, source, d + 1);
    }
}

void dfs2(ll source, ll parent, ll d){
    if (dM <= d) dM = d, v2 = source;
    rep(i, tree[source].size()){
        ll target = tree[source][i].to;
        if (target == parent) continue;
        dfs2(target, source, d + 1);
    }
}

dfs(0, -1, 0);
dfs2(vM, -1, 0);
prl2(vM + 1, v2 + 1); // the two edges of tree


const ll N_VERTEX = 310;
ll a, b, t;
vector<vl> dist;

void warshall_floyd(ll n){
    // rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(k, n) rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][k] +
dist[k][j]);
}

int main(void){
    in2(n, m);
    rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        dist[a][b] = t;
        dist[b][a] = t;
    }

    warshall_floyd(n);
}

vl dist, vertex_pre;

void dijkstra(ll start, ll n) {
    priority_queue<pl, vector<pl>, greater<pl>> edge_costs;
    dist = vl(n, INF); // vertex_pre = vl(n, -1);
    dist[start] = 0;
    edge_costs.push(pl(0, start));

    while (!edge_costs.empty()) {
        pl edge_cost = edge_costs.top();
        edge_costs.pop();
        ll idx = edge_cost.second;
        ll cost = edge_cost.first;
        if (dist[idx] < cost) continue;

        rep(i, graph[idx].size()){
            edge e = graph[idx][i];
            if (dist[e.to] > dist[idx] + e.cost){
                dist[e.to] = dist[idx] + e.cost;
                // vertex_pre[e.to] = idx;
                edge_costs.push(pl(dist[e.to], e.to));
            }
        }
    }
}

vl get_predecessor(ll g){
    vl path;
    for (; g != -1; g = vertex_pre[g]) path.pb(g);

    reverse(all(path));
    return path;
}

int main(void){
    in2(n, m);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        G[a].pb({b, t});
        G[b].pb({a, t});
    }

    dijkstra(0, n);
}


# ABC061D

bool find_negative_cycle(ll goal){
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[e.to] != INF && dist[e.to] > dist[v] + e.cost){
            dist[e.to] = -INF;
            if (goal == -1) return true;
            else if (goal == e.to) return true;
        }
    }
    return false;
}

bool bellman_ford(ll start, ll n, ll goal){
    // if there is a closed circuit, it returns false. (when goal == -1)
    // if the distance to goal cannot be obtained, it returns false (when goal
!= -1) fill(dist, dist + n, INF); dist[start] = 0; rep(i, n) rep(v, n) rep(k,
graph[v].size()){ edge e = graph[v][k]; if (dist[v] != INF && dist[e.to] >
dist[v] + e.cost) dist[e.to] = dist[v] + e.cost;
    }

    if (find_negative_cycle(goal)) return false;
    return true;
}

*/

/*
    # 1. The usage of map pair
    map<pl, ll> cnt;
    cnt[{i, j}] = 0;
    items(kv, cnt){
        prl2(kv.first, kv.second);
    }

    # 2. The usage of next_permutation and combination (factorial search)
    ll a[8];
    rep(i, 8) a[i] = i;
    sort(a, a + 8);
    do{
    }while(next_permutation(a, a+n));

    // here, combination
    ll n, r;
    ll i1, i2, ..., ir;
    For(i1, n - r, n) For(i2, n - r - 1, i1) ... For(ir, n - 2 * r + 1, irr){
   process;}

    # 3. bit search
    ll n;
    in1(n);
    const ll base = 3;
    ll upper = power_normal(base, n);

    rep(i, upper){
        ll tmp = i;
        rep(j, n){
            rep(k, base) if (tmp % base == k) prl(k);
            tmp /= base;
        }
    }

    # 4. imos method
    // used when we would like to count the number which
    // shows how many times the numbers between l and r belongs to smt.
    // This method is composed of three process.
    ll n, m, s[MAX_M], l, r;
    in2(n, m);
    rep(i, m) s[i] = 0;

    // 1st step
    rep(i, n){
        in3(l, r, c);
        l--; r--; // if l starts from 1.
        s[l] += c; s[r + 1] -= c;
    }

    // 2nd step
    rep(i, m - 1) s[i + 1] += s[i];
    // 3rd step: judgement...

    #5. shakutori method (syakutori, two pointers technique)
    // 1. strech right side while the condition is met.
    // 2. renew the answer
    // 3. increments left side
    // 4. Back to 1. (l <= r must be satisfied all the time.)

    ll l = 0; ll r = 0;
    while (l < n){
        r = max(r, l);
        if (l == r) r++;
        while(r < n && cond) r++;
        answer += r - l; l++;
    }
    prl(answer);

    #6. priority queue

    pq q;
    ll answer = 0;
    ll v;

    rep(i, n) q.push(a[i]);
    rep(i, m){
        v = q.top(); q.pop(); // get the top value and dump the value from queue
        v /= 2; q.push(v); // add the new value
    }

    while(!q.empty()){
        answer += q.top();
        q.pop();
    }

    #7. The shortest path (distance) between the k-th edge and another edge
   (Tree) Graph tree[MAX_N]; ll depth[MAX_N];

    void dfs(ll source, ll parent, ll all_cost){
        depth[source] = all_cost;
        items(e, tree[source]){
            if (e.to == parent) continue;
            dfs(e.to, source, all_cost + e.cost);
        }
    }

    ll n, k, a, b, c;

    in2(n, k);
    rep(i, n - 1){
        in3(a, b, c);
        a--; b--;
        tree[a].pb({b, c});
        tree[b].pb({a, c});
    }

    k--;
    dfs(k, -1, 0);

    #10. Visiting Subtree using recurrent function (ABC138D)

    Graph tree[MAX_N];
    ll c[MAX_N];
    bool visited[MAX_N];

    void dfs(ll source, ll parent, ll val){
        visited[source] = true;
        c[source] += val;
        rep(i, tree[source].size()){
            rep(i, m){ll res = n % match[i].e1;}
            ll vertex = tree[source][i].to;
            if (vertex == parent) continue;
            dfs(vertex, source, c[source]);
        }
    }

    #11. bfs ABC146D, ABC007C
    1. first create a tree.
    2. start searching from a node.
    3. do some processes and push nodes connected with a given target node in
   BFS.
    4. repeat a series of procedure until queue is empty.

    queue<pl> q;

    void bfs(ll source, ll parents){
        ll n_edge = graph[source].size();
        if (parents != -1) dist[source] = min(dist[source], dist[parents] + 1);
        if (visited[source]) return;
        visited[source] = true;

        rep(idx, n_edge){
            ll target = graph[source][idx].to;
            if (target == parents) continue;
            q.push(mp(target, source));
        }
    }

    q.push(mp(sg.e1, -1));
    while(!q.empty()){
        pl source = q.front(); q.pop();
        bfs(source.e1, source.e2);
    }

    #12. grid to distance matrix (dx, dy)
    ll w, h;
    ll pos_to_idx(ll x, ll y){
        return y * w + x;
    }

    pl idx_to_pos(ll idx){
        return mp(idx % w, idx / w);
    }

    rep(y, h){
        in1(s);
        rep(x, w){
            if (s[x] == '#') wall[x][y] = true;
            else wall[x][y] = false;
        }
    }

    rep(i1, h * w)rep(i2, h * w) dist[i1][i2] = INF * (i1 != i2);

    rep(x, w)rep(y, h){
        ll idx1 = pos_to_idx(x, y); ll idx2;
        if (wall[x][y]) continue;
        if (x != 0 && !wall[x - 1][y]){
            idx2 = pos_to_idx(x - 1, y);
            // if warshall floyd
            dist[idx1][idx2] = 1;
            // if dijkstra
            // graph[idx1].pb({idx2, 1});
        }
        if (x != w - 1 && !wall[x + 1][y]){
            idx2 = pos_to_idx(x + 1, y);
            dist[idx1][idx2] = 1;
            // graph[idx1].pb({idx2, 1});
        }
        if (y != 0 && !wall[x][y - 1]){
            idx2 = pos_to_idx(x, y - 1);
            dist[idx1][idx2] = 1;
            // graph[idx1].pb({idx2, 1});
        }
        if (y != h - 1 && !wall[x][y + 1]){
            idx2 = pos_to_idx(x, y + 1);
            dist[idx1][idx2] = 1;
            // graph[idx1].pb({idx2, 1});
        }
    }

    # Cumulative sum (2 dimension)

    ll func(ll x, ll y, ll dx, ll dy){
        if (x + dx > w || y + dy > h) return - INF;
        ll val = cum[x + dx][y + dy];
        val += cum[x][y];
        val -= cum[x][y + dy];
        val -= cum[x + dx][y];
        return val;
    }

    rep(x, w + 1) cum[x][0] = 0;
    rep(y, h + 1) cum[0][y] = 0;
    rep(y, h + 1) rep(x, w)
        cum[x + 1][y + 1] = cum[x][y + 1] + vs[x][y];

    rep(x, w + 1) rep(y, h)
        cum[x][y + 1] += cum[x][y];

    # DEBUG_MATRIX

    rep(i, n){
        rep(j, n) prs(a[i][j]);
        prl("");
    }

*/

/*
# the operators regarding bit
& (AND), | (OR), ^ (XOR)
- (REVERSE), >> (SMALLER SHIFT)
<< (BIGGER SHIFT)


     x1: 0000 0001 0010 0101 0110 0111 0111
     x2: xxxx 0001 0011 0100 0101 1000 0110
x1 & x2: 0000 0001 0010 0100 0100 0000 0110

      x: 1001 1010  1100 1011 1101  1111
x & - x: 0001 0010  0100 0001 0001  0001
    sum: 1010 1100 10000 1100 1100 10000

x << y is x * 2 ** y
x >> y is rep(i, y) x = x // 2

####### Attention #######
1 << i and 1LL << i is different. If programs show WA, try switch to this one.
Let S be a bit sequence and i be a non-negative integer
S & (1 << i) -> if true, i in S
S | (1 << i) -> S union {i}
S & -(1 << i) -> S - {i}
__builtin_popcount(S) -> the number of elements in S
S = 0 -> S is an empty set
__builtin_popcountl(i) -> the number of 1 in binary
S = (1 << n) - 1 -> S includes all the elements up to the n-th

#Conditional Operator
condition ? true : false;

#iterator
type declaration: auto
value reference: *itr
increment: itr++
decrement: itr--
substitution of value: *itr = smt

# inclusion-exclusion principle
|U[i = 1 to n] Ai| = sum[i = 1 to n] |Ai| - sum[i < j]|Ai ^ Aj| + ... + (-1)^(n
- 1) |^[i = 1 to n]Ai|
*/

const ll MAX_N = 200005;
bool okay = false;
ll answer = 0;
ll n, m, k;

// ABC160F is one example
// Modify the functions evaluate and merge based on given problems
struct Merger {
  using type = ll;

  static type merge(const vector<type> &value) {
    // merge the result below the source node
    // each value is from each child node of the source node
    // value[i] := f(child i)
    // Here, we would like to obtain f(source) using f(child i) (i = 0, 1, ...,
    // n_children)
    ll ret = 1;
    for (auto &&v : value)
      ret += v;
    return ret;
  }

  static vector<type> evaluate(const vector<type> &value) {
    // value[i] := f(source -> child i)
    // we would like to obtain f(child i -> source)
    // child j (j != i) is the grandchildren of child i
    // represent f(child i -> source) using f(source -> j) (j != i)
    // L[i + 1] := the result using f(source -> k) (k = 0, 1, ..., i)
    // R[i] := the result using f(source -> k) (k = i, i + 1, ..., n_children)

    const ll n_children = value.size();
    vl L(n_children + 1, 0), R(n_children + 1, 0);
    rep(i, n_children) L[i + 1] = L[i] + value[i];
    revrep(i, n_children) R[i] = R[i + 1] + value[i];

    vl ret(n_children);
    rep(i, n_children) ret[i] = L[i] + R[i + 1] + 1;

    return ret;
  }
};

ll memo[MAX_N];

void solve() {
  RerootingTreeDP<Merger> dp(n);
  rep(i, n - 1) {
    ll a, b;
    in2(a, b);
    a--, b--;
    dp.add_edge(a, b, 1);
  }

  memo[0] = 1;
  rep(i, n + 3) memo[i + 1] = (memo[i] * 2) % MOD;
  dp.dp();
  rep(i, n) { // dp.tree[i] := the number of children
    (answer += memo[n - 1] - 1) %=
        MOD;                    // all the cases except all of them are white
    for (auto &&e : dp.tree[i]) // the case where the i-th vertex is not a hole
      (answer += MOD - (memo[e.value] - 1)) %= MOD;
  }

  ll x_inv = modinv(power_mod(2, n, MOD), MOD);
  ll y = answer;

  prl((y * x_inv) % MOD);
}

int main(void) {
  in1(n);
  assert(n <= 9);
  solve();
  return 0;
}


Fixed submission:
#include <algorithm>
#include <bits/stdc++.h>
#include <cassert>
#include <complex>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <iostream>
#include <list>
#include <map>
#include <queue>
#include <set>
#include <sstream>
#include <stack>
#include <stdio.h>
#include <string>
#include <vector>
using namespace std;
typedef long long ll;
typedef long double ld;
using edge = struct {
  ll to;
  ll cost;
};
struct point {
  ll x;
  ll y;
  bool operator<(const point &p) const {
    if (x == p.x)
      return y < p.y;
    return x < p.x;
  }
};
struct undirected_edge {
  ll from;
  ll to;
  ll cost;
  bool operator<(const undirected_edge &ue) const { return cost < ue.cost; }
};

typedef string str;
typedef std::pair<ll, ll> pl;
typedef std::pair<ll, pl> pl3;
typedef std::map<string, ll> msl;
typedef std::map<char, ll> mcl;
typedef std::map<ll, ll> mll;
typedef std::vector<ll> vl;
typedef std::vector<pl> vpl;
typedef std::vector<point> points;
typedef std::vector<pl3> vpl3;
typedef std::priority_queue<ll> pq;
typedef std::priority_queue<ll, vl, greater<ll>>
    pql; // priority queue taking from the lower value.
typedef std::vector<edge> Graph;
const ll N_DIGITS = 60;
const ll MOD = 1e9 + 7;
// const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-9;
const long double PI = 3.14159265358979323846;
vpl dirs = {
    {-1, 0}, {1, 0},  {0, 1},   {0, -1}, // four directions
    {1, 1},  {-1, 1}, {-1, -1}, {1, -1}, // diagonal
    {0, 0}                               // self
};

template <typename A, typename B> string to_string(pair<A, B> p);

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);

string to_string(const string &s) { return '"' + s + '"'; }

string to_string(const char *s) { return to_string((string)s); }

string to_string(bool b) { return (b ? "true" : "false"); }

string to_string(vector<bool> v) {
  bool first = true;
  string res = "{";
  for (int i = 0; i < static_cast<int>(v.size()); i++) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(v[i]);
  }
  res += "}";
  return res;
}

template <size_t N> string to_string(bitset<N> v) {
  string res = "";
  for (size_t i = 0; i < N; i++) {
    res += static_cast<char>('0' + v[i]);
  }
  return res;
}

template <typename A> string to_string(A v) {
  bool first = true;
  string res = "{";
  for (const auto &x : v) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(x);
  }
  res += "}";
  return res;
}

template <typename A, typename B> string to_string(pair<A, B> p) {
  return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " +
         to_string(get<2>(p)) + ")";
}

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " +
         to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}

void debug_out() { cerr << endl; }

template <typename Head, typename... Tail> void debug_out(Head H, Tail... T) {
  cerr << " " << to_string(H);
  debug_out(T...);
}

#define rep(i, n) for (ll(i) = 0; (i) < (n); (i)++)
#define revrep(i, n) for (ll(i) = n - 1; (i) >= 0; (i)--)
#define For(i, a, b) for (ll(i) = (a); (i) < (b); (i)++)
#define revFor(i, b, a) for (ll(i) = (b)-1; (i) >= (a); (i)--)
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define isUpper(c) ('a' - c > 0)
#define isNum(c) (0 <= (c) - '0' && (c) - '0' <= 9)
#define toLower(c) char((c) + 0x20)
#define toUpper(c) char((c)-0x20)
#define pb push_back
#define mp make_pair
#define pr(a) cout << (a)
#define prl(a) cout << (a) << endl
#define prl2(a, b) cout << (a) << " " << (b) << endl
#define prl3(a, b, c) cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a, b, c, d)                                                       \
  cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) cout << (a) << " "
#define prs2(a, b) cout << (a) << " " << (b) << " "
#define prs3(a, b, c) cout << (a) << " " << (b) << " " << (c) << " "
#define prs4(a, b, c, d)                                                       \
  cout << (a) << " " << (b) << " " << (c) << " " << (d) << " "
#define yn(condition)                                                          \
  if ((condition))                                                             \
    prl("Yes");                                                                \
  else                                                                         \
    prl("No");
#define YN(condition)                                                          \
  if ((condition))                                                             \
    prl("YES");                                                                \
  else                                                                         \
    prl("NO");
#define in1(a) cin >> (a)
#define in2(a, b) cin >> (a) >> (b)
#define in3(a, b, c) cin >> (a) >> (b) >> (c)
#define in4(a, b, c, d) cin >> (a) >> (b) >> (c) >> (d)
#define e1 first
#define e2 second
#define ctol(c) ll((c)) - ll('0')
#define ltos(n) to_string((n))
#define items(kv, v) for (auto &(kv) : (v))
#define ndig(N, n) ctol(ll(ltos((N))[ll(ltos((N)).length()) - (n)]))
#define rsort(a, n) sort(a, a + n, greater<>())
#define Forchar(c, a, z) for (char(c) = (a); (c) <= (z); (c)++)
#define cntchar(s, c) count(all((s)), c)
#define substring(s, start, end) s.substr((start), (end) - (start) + 1)
#define prl_nd(num, digits)                                                    \
  cout << fixed << setprecision(digits) << (num) << endl;
#define XOR(a, b) (a) ^ (b)
#define prl_time(s)                                                            \
  prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))
#define DEBUG(x) cerr << #x << ": " << (x) << endl
#define DEBUG_VEC(v)                                                           \
  cerr << #v << ": ";                                                          \
  rep(__i, (v).size()) cerr << ((v)[__i]) << ", ";                             \
  cerr << endl
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)

template <class T> inline bool chmax(T &a, T b) {
  if (a < b) {
    a = b;
    return true;
  }
  return false;
}

template <class T> inline bool chmin(T &a, T b) {
  if (a > b) {
    a = b;
    return true;
  }
  return false;
}

struct MaxFlow {
  struct F_edge {
    ll to, rev, capacity;
    F_edge(ll to, ll rev, ll capacity) : to(to), rev(rev), capacity(capacity) {}
  };

  typedef vector<F_edge> F_edges;
  vector<F_edges> graph;
  ll n_vertex;
  // level is the shortest path to get a given node from the source node.
  vl level, iter;

  MaxFlow(ll n_vertex) : n_vertex(n_vertex) { graph.resize(n_vertex); }

  void add_edge(ll from, ll to, ll capacity) {
    graph[from].pb({to, ll(graph[to].size()), capacity});
    graph[to].pb({from, ll(graph[from].size()) - 1, 0});
  }

  void bfs(ll source) {
    level = vl(n_vertex, -1);
    level[source] = 0;
    queue<ll> q;
    q.push(source);
    while (!q.empty()) {
      ll vertex = q.front();
      q.pop();
      rep(i, graph[vertex].size()) {
        ll target = graph[vertex][i].to;
        ll cap_target = graph[vertex][i].capacity;
        // if the flow can be into the target node, implement below.
        if (cap_target > 0 && level[target] < 0) {
          level[target] = level[vertex] + 1;
          q.push(target);
        }
      }
    }
  }

  ll dfs(ll vertex, ll sink, ll flow) {
    if (vertex == sink)
      return flow;
    for (ll &i = iter[vertex]; i < graph[vertex].size(); i++) {
      ll target = graph[vertex][i].to;
      ll cap_target = graph[vertex][i].capacity;
      ll rev_target = graph[vertex][i].rev;
      // if capasitiy is not full yet and target is farther,
      // then assign current flow
      if (cap_target > 0 && level[vertex] < level[target]) {
        ll d = dfs(target, sink, min(cap_target, flow));
        if (d > 0) { // if the flow successfully reaches the sink, reduce the
                     // flow from the capacity
          graph[vertex][i].capacity -= d;
          graph[target][rev_target].capacity += d;
          return d;
        }
      }
    }
    return 0;
  }

  ll dinic(ll source, ll sink) {
    ll flow = 0;
    while (true) {
      bfs(source);
      // if there is no path leading to the sink, the maximum flow is 0.
      if (level[sink] < 0)
        return flow;
      iter = vl(n_vertex, 0);
      ll f;
      while ((f = dfs(source, sink, INF)) > 0)
        flow += f;
    }
  }
};

class UnionFind {
  vl parents, set_size;

public:
  UnionFind() {}
  UnionFind(ll n) {
    parents = set_size = vl(n);
    rep(i, n) {
      parents[i] = i;
      set_size[i] = 1LL;
    }
  }

  ll root_find(ll x) {
    if (parents[x] == x)
      return x;
    return parents[x] = root_find(parents[x]);
  }

  void unite(ll x, ll y) {
    // priority for x is larger than that of y
    x = root_find(x);
    y = root_find(y);
    if (x == y)
      return;
    parents[y] = x, set_size[x] += set_size[y];
  }

  bool is_same(ll x, ll y) { // connected or not
    return root_find(x) == root_find(y);
  }

  ll size(ll x) { return set_size[root_find(x)]; }
};

struct Doubling { // ABC167D
  ll n;
  ll sz;
  vector<vl> next;
  /*
      next[k + 1][i] := next[k][next[k][i]]
      next[0][i] := edge[i]
      e.g. a0, a1, ..., an-1 / 0 <= ai <= n - 1
      a0 -> a[a0] -> a[a[a0]] -> ... -> a[a[...[a[0]]]] (m times)
      Let the function repeatedly input a[i] m times be f[m](a[i])

      - get(i, x) returns f[x](a[i])
      - lower_bound(i, j) returns minimum x which satisfies f[x](a[i]) >= j.
                                  if not possible returns n.

  */

  // edge[i]: the step size for one iteration
  Doubling(vl &edge) : n(edge.size()), sz(62) {
    next.resize(sz, vl(n, -1));
    rep(i, n) next[0][i] = edge[i];
    rep(k, sz - 1) rep(i, n) next[k + 1][i] = next[k][next[k][i]];
  }

  ll get(ll i, ll x) {
    ll ret = i;
    rep(bit, sz) {
      if (!(x >> bit & 1))
        continue;
      ret = next[bit][ret];
    }
    return ret;
  }

  ll lower_bound(ll i, ll j) {
    ll cur = i, acc = 0;
    revrep(wid, sz) {
      if (next[wid][cur] < j) {
        acc += 1LL << wid;
        cur = next[wid][cur];
      }
    }
    return min(n, acc + 1);
  }
};

/*
class LCA{
public:
    ll N, logN;
    vl depth, len;
    Graph tree[200005]; // global declaration later.
    vector<vl> parents;

    LCA(ll n){
        N = n;
        logN = 0;
        while (N > (1LL << logN)) logN++;
        depth = vl(N); len = vl(N);
        parents = vector<vl>(logN, vl(N));
        init(0, -1, 0, 0);
        build();
    }

    void init(ll source, ll parent, ll d, ll l){
        depth[source] = d;
        parents[0][source] = parent;
        len[source] = l;
        rep(i, tree[source].size()){
            ll target = tree[source][i].to;
            ll cost = tree[source][i].cost;
            if (target == parent) continue;
            init(target, source, d + 1, cost + l);
        }
    }

    void build(){
        rep(k, logN - 1) rep(n, N){
            // if there is no parent, -1.
            // otherwise, the parent of the parent is the parent.
            if (parents[k][n] < 0) parents[k + 1][n] = -1;
            else parents[k + 1][n] = parents[k][parents[k][n]];
        }
    }

    ll query(ll u, ll v){
        if (depth[u] > depth[v]) swap(u, v);
        rep(k, logN) if ((depth[v] - depth[u]) >> k & 1) v = parents[k][v];
        if (u == v) return u;

        revrep(k, logN){
            if (parents[k][u] != parents[k][v]){
                u = parents[k][u]; v = parents[k][v];
            }
        }
        return parents[0][u];
    }

    ll distance(ll u, ll v){
        ll w = query(u, v);
        return len[u] + len[v] - 2 * len[w];
    }

};
*/

struct BIT {
  ll n;
  vl dat;
  BIT(ll n, ll ini = 0) : dat(n + 1, ini), n(n){};

  //       x: 1001 1010  1100 1011 1101  1111
  // x & - x: 0001 0010  0100 0001 0001  0001
  //      ->: 1010 1100 10000 1100 1100 10000

  ll update_func(ll val, ll d) {
    // if maximum -> max(val, dat)
    // return max(val, dat);
    // if cumulative sum
    return val + d;
  }

  ll query(ll i) {
    /*
    v[0] + v[1] + ... + v[i]
    e.g.) i = 10101
    itr1. 10101 -> 10100
    itr2. 10100 -> 10000
    itr3. 10000 -> 00000 (break)
    */
    ll ret = 0;
    for (ll j = i; j >= 0; j = (j & (j + 1)) - 1) {
      ret = update_func(ret, dat[j]);
    }
    return ret;
  }

  ll query(ll l, ll r) { return query(r) - query(l); }

  ll lower_bound(ll key) {
    // v[0] + v[1] + ... + v[left - 1] < key <= v[0] + v[1] + ... + v[left]
    if (key <= 0)
      return 0;
    ll left = 0, right = 1;
    while (right <= n)
      right *= 2;

    for (ll i = right; i > 0; i /= 2) {
      if (left + i <= n && dat[left + i - 1] < key) {
        key -= dat[left + i - 1];
        left += i;
      }
    }
    return left;
  }

  void update(ll i, ll val) {
    /*
    e.g.) i = 10101, n = 11111
    itr1. i: 10101, i+1: 10110 -> 10111
    itr2. i: 10111, i+1: 11000 -> 11111 (break)
    */
    if (i < 0)
      return;
    for (ll j = i; j < n; j |= j + 1) {
      dat[j] = update_func(val, dat[j]);
    }
  }
};

struct SegmentTree {
  ll n, ini, minimize;
  vl dat;

  // when seeking minimum
  // ini = INF
  // when seeking maximum
  // ini = -INF
  SegmentTree(ll n_, bool minimize_ = true) {
    n = 1;
    minimize = minimize_;
    if (minimize)
      ini = INF;
    else
      ini = -INF;
    while (n < n_)
      n *= 2;
    dat.resize(2 * n - 1);
    rep(i, 2 * n - 1) dat[i] = ini;
  };

  void update(ll idx, ll val) {
    idx += n - 1;
    if (minimize && dat[idx] <= val)
      return;
    if (!minimize && dat[idx] >= val)
      return;

    dat[idx] = val;
    while (idx > 0) {
      idx = (idx - 1) / 2;
      // when seeking minimum
      if (minimize)
        dat[idx] = min(dat[idx * 2 + 1], dat[idx * 2 + 2]);
      // when seeking maximum
      else
        dat[idx] = max(dat[idx * 2 + 1], dat[idx * 2 + 2]);
    }
  }

  ll query(ll l, ll r) {
    // ### NOTE ###
    // the range is [l, r]
    // l, l + 1, ..., r
    r++; // to adjust to this method
    return query_segment(l, r, 0, 0, n);
  }

  ll query_segment(ll a, ll b, ll idx, ll l, ll r) {
    assert(a < b);

    if (r <= a || b <= l)
      return ini;
    if (a <= l && r <= b)
      return dat[idx];
    else {
      ll seg1 = query_segment(a, b, idx * 2 + 1, l, (l + r) / 2);
      ll seg2 = query_segment(a, b, idx * 2 + 2, (l + r) / 2, r);
      // when seeking minimum
      if (minimize)
        return min(seg1, seg2);
      // when seeking maximum
      else
        return max(seg1, seg2);
    }
  }
};

template <class Target> class RerootingTreeDP {
public:
  using T = typename Target::type;
  struct DP_edge {
    ll to, rev; // rev is the index to trace the source node.
    T value;    // objective value
  };

private:
  ll n;
  void dfs_fwd(ll source, ll parent) {
    ll par_idx = -1;
    vector<T> values;
    rep(i, tree[source].size()) {
      const DP_edge &e = tree[source][i];
      if (e.to == parent) {
        par_idx = i;
        continue;
      }
      dfs_fwd(e.to, source);
      values.pb(e.value);
    }

    // If the parent != -1, update the value on edge from parent to source
    if (par_idx != -1) {
      ll src_idx = tree[source][par_idx].rev;
      // update values on the edge from parent to source
      tree[parent][src_idx].value = Target::merge(values);
    }
  }

  void dfs_bwd(ll source, ll parent) {
    vector<T> values;
    for (auto &&e : tree[source])
      values.pb(e.value);
    values = Target::evaluate(values);

    rep(i, tree[source].size()) {
      const DP_edge &e = tree[source][i];
      if (e.to == parent)
        continue;
      // tree[e.to][e.rev]: e.to -> source
      tree[e.to][e.rev].value = values[i];
      dfs_bwd(e.to, source);
    }
  }

public:
  UnionFind uf;
  vector<vector<DP_edge>> tree;
  RerootingTreeDP(ll n) : n(n), uf(n), tree(n) {}

  void add_edge(ll u, ll v, T val) {
    assert(!uf.is_same(u, v));
    tree[u].pb({v, ll(tree[v].size()), val});
    tree[v].pb({u, ll(tree[u].size()) - 1, val});
    uf.unite(u, v);
  }

  void dp() {
    vector<bool> visited(n, false);
    rep(i, n) {
      if (visited[uf.root_find(i)])
        continue;
      dfs_fwd(i, -1);
      visited[uf.root_find(i)] = true;
    }

    visited.assign(n, false);
    rep(i, n) {
      if (visited[uf.root_find(i)])
        continue;
      dfs_bwd(i, -1);
      visited[uf.root_find(i)] = true;
    }
  }

  ll size() const { return tree.size(); }
};

struct StronglyConnectedComponents {
  ll n;
  vector<vl> graph, graph_rev, dag, cmp;
  vl order, visited, cmp_idx;

  StronglyConnectedComponents() {}
  StronglyConnectedComponents(ll sz)
      : n(sz), graph(sz), graph_rev(sz), visited(sz), cmp_idx(sz) {}

  void add_edge(ll from, ll to) {
    graph[from].pb(to);
    graph_rev[to].pb(from);
  }

  void input(ll m, ll offset = -1) {
    ll a, b;
    rep(i, m) {
      in2(a, b);
      add_edge(a + offset, b + offset);
    }
  }

  ll operator[](ll k) { return cmp_idx[k]; }

  void dfs_fwd(ll source) {
    visited[source] = 1;
    rep(i, graph[source].size()) {
      ll target = graph[source][i];
      if (!visited[target])
        dfs_fwd(target);
    }
    order.pb(source);
  }

  void dfs_bwd(ll source, ll num) {
    visited[source] = 1, cmp_idx[source] = num;

    cmp[num].pb(source);
    rep(i, graph_rev[source].size()) {
      ll target = graph_rev[source][i];
      if (!visited[target])
        dfs_bwd(target, num);
    }
  }

  ll build() {
    fill(all(visited), 0);
    order.clear();
    rep(i, n) if (!visited[i]) dfs_fwd(i);

    fill(all(visited), 0);
    ll num = 0;
    revrep(i, order.size()) {
      if (!visited[order[i]]) {
        dag.pb(vl());
        cmp.pb(vl());
        dfs_bwd(order[i], num++);
      }
    }

    rep(i, n) for (ll to : graph[i]) if (cmp_idx[i] != cmp_idx[to])
        dag[cmp_idx[i]]
            .pb(cmp_idx[to]);

    rep(i, num) {
      sort(all(dag[i]));
      dag[i].erase(unique(all(dag[i])), dag[i].end());
    }
    return num;
  }
};

ll gcd(ll m, ll n) {
  ll a = max(m, n);
  ll b = min(m, n);
  while (b != 1 && b != 0) {
    a %= b;
    swap(a, b);
  }
  return b == 1 ? 1 : a;
}

ll lcm(ll m, ll n) { return m / gcd(m, n) * n; }

ll power_mod(ll a, ll power, ll mod) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = (value * a) % mod;
    a = (a * a) % mod;
    power = power >> 1;
  }
  return value % mod;
}

ll modinv(ll a, ll mod) { return power_mod(a, mod - 2, mod); }

ll power_normal(ll a, ll power) {
  ll value = 1;
  while (power != 0) {
    if (power & 1)
      value = value * a;
    a = a * a;
    power = power >> 1;
  }
  return value;
}

ll comb_memo[55][55];

ll pascal_triangle(ll n) {
  comb_memo[0][0] = 1;
  For(i, 1, n + 1) rep(j, i + 1) {
    comb_memo[i][j] += comb_memo[i - 1][j];
    if (j > 0)
      comb_memo[i][j] += comb_memo[i - 1][j - 1];
  }
}

ll combination(ll n, ll r, ll mod) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    ll num = (n - i) % mod, den = (i + 1) % mod;
    (numerator *= num) %= mod;
    (denomenator *= modinv(den, mod)) %= mod;
  }

  return (numerator * denomenator) % mod;
}

ll combination_no_mod(ll n, ll r) {
  if (n == r && n == 0)
    return 1;
  else if (n <= 0 || r < 0 || r > n)
    return 0;
  ll numerator = 1;
  ll denomenator = 1;
  for (ll i = 0; i < r; i++) {
    numerator *= n - i;
    denomenator *= i + 1;
    ll g = gcd(numerator, denomenator);
    numerator /= g;
    denomenator /= g;
  }

  return numerator;
}

ld log_combination(ll n, ll r) {
  if (n == r && n == 0)
    return 0;
  else if (n <= 0 || r < 0 || r > n)
    return -INF;
  ld val = 0;
  for (ll i = 0; i < r; i++) {
    val += log(n - i);
    val -= log(i + 1);
  }

  return val;
}

ll bin_search(ll key, ll A[], ll left, ll right) {
  // return the index idx where A[idx] = key.
  // A[left] is start and A[right] is end..
  // In other words, A[right], not A[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (A[mid] == key)
      return mid;
    else if (A[mid] > key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid + 1;
  }
  return -1;
}

/*
ll bin_search_temp(ll left, ll right, callable judge){
    while(right > left){
        // when seeking lower bound
        ll mid = (right + left) / 2;
        if (judge(mid)) right = mid;
        else left = mid + 1;
        // when seeking upper bound
        ll mid = (right + left + 1) / 2;
        if (judge(mid)) left = mid;
        else right = mid - 1;
    }

    return right;
}

trinary_search

// Care the value EPS!!! Compare to the condition
ld left = 0; ld right = p;
while(abs(right - left) > EPS){
    ld left2 = (2 * left + right) / 3.0;
    ld right2 = (left + 2 * right) / 3.0;
    ld f1 = tak_func(left2);
    ld f2 = tak_func(right2);
    if (f1 <= f2) right = right2;
    else if (f2 <= f1) left = left2;
}

*/

str bin_expression(ll n, ll dig) {
  str s = "";
  rep(i, dig) {
    s += ltos(n % 2);
    n /= 2;
  }
  reverse(all(s));
  return s;
}

ll lower_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] >= key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] < key && key <= A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] < key && key <= A[mid])
      return mid;
    else if (A[mid - 1] >= key)
      right = mid - 1;
    else if (A[mid] < key)
      left = mid;
  }

  return -1; // all the elements < key
}

ll inf_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value right instead of -1.
  if (A[left] > key)
    return left;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == left) {
      if (A[left] <= key && key < A[left + 1])
        return left + 1;
      else
        return -1;
    }
    if (A[mid - 1] <= key && key < A[mid])
      return mid;
    else if (A[mid - 1] > key)
      right = mid - 1;
    else if (A[mid] <= key)
      left = mid;
  }

  return -1; // all the elements <= key
}

ll upper_bound_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] <= key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] <= key && key < A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] <= key && key < A[mid + 1])
      return mid;
    else if (A[mid] > key)
      right = mid;
    else if (A[mid + 1] <= key)
      left = mid + 1;
  }

  return -1; // all the elements > key
}

ll sup_bin_search_temp(ll key, ll A[], ll left, ll right) {
  // Pay attention to the return value -1.
  // For example, sometimes you may want the value left instead of -1.
  if (A[right] < key)
    return right;
  ll mid = left + (right - left) / 2;

  while (right >= left) {
    mid = left + (right - left) / 2;
    if (mid == right) {
      if (A[right - 1] < key && key <= A[right])
        return right - 1;
      else
        return -1;
    }
    if (A[mid] < key && key <= A[mid + 1])
      return mid;
    else if (A[mid] >= key)
      right = mid;
    else if (A[mid + 1] < key)
      left = mid + 1;
  }

  return -1; // all the elements >= key
}

ll bin_search_vector(ll key, vl v, ll left, ll right) {
  // return the index idx where v[idx] = key.
  // v[left] is start and v[right] is end..
  // In other words, v[right], not v[right - 1], must be defined.
  while (right >= left) {
    ll mid = left + (right - left) / 2;
    if (v[mid] == key)
      return mid;
    else if (v[mid] > key)
      right = mid - 1;
    else if (v[mid] < key)
      left = mid + 1;
  }
  return -1;
}

ll lower_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] < key <= v[N]
  // N == -1 if all the elements < key
  return lower_bound(all(v), key) - v.begin();
}

ll inf_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N - 1] <= key < v[N] <= key + 1
  // N == -1 if all the elements <= key
  return lower_bound(all(v), key + 1) - v.begin();
}

ll upper_bound_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key < v[N + 1]
  // N == -1 if all the elements > key
  return upper_bound(all(v), key) - v.begin(); // (- 1)
}

ll sup_bin_search_vector(ll key, vl v) {
  // the return value N satisfies
  // v[N] <= key - 1 < key <= v[N + 1]
  // N == -1 if all the elements >= key
  return upper_bound(all(v), key - 1) - v.begin() - 1;
}

ll fact(ll n) {
  if (n == 0)
    return 1;
  return n * fact(n - 1);
}

ll fact_mod(ll n, ll mod) {
  if (n == 0)
    return 1;
  return n * fact_mod(n - 1, mod);
}

/*
ll facts[6000], facts_inv[6000], inv[6000];

void fact_memo(ll n, ll mod){
    facts[0] = facts[1] = 1;
    facts_inv[0] = facts_inv[1] = 1;
    inv[1] = 1;

    For(i, 2, n + 1){
        facts[i] = (i * facts[i - 1]) % mod;
        inv[i] = mod - inv[mod % i] * (mod / i) % mod;
        facts_inv[i] = facts_inv[i - 1] * inv[i] % mod;
    }
}

ll nCk(ll n, ll r){
    if (n == r && n == 0) return 1;
    else if (n <= 0 || r < 0 || r > n) return 0;
    ll val = (facts[n] * facts_inv[n - r]) % MOD;
    val *= facts_inv[r];
    return val % MOD;
}
*/

bool is_prime(ll n) {
  if (n <= 1)
    return false;
  for (ll i = 2; i * i <= n; i++) {
    if (n % i == 0)
      return false;
  }
  return true;
}

ll bool_sum(ll a1, ll a2) {
  if (a1 == 1 || a2 == 1)
    return 1;
  return 0;
}

mll prime_factorization(ll n) {
  ll i = 2;
  mll table;
  while (i * i <= n) {
    while (n % i == 0) {
      table[i]++;
      n /= i;
    }
    i++;
  }
  if (n > 1)
    table[n] = 1;
  return table;
}

vl divisor_table(ll n) {
  vl table;
  ll i = 1;
  while (i * i <= n) {
    if (n % i == 0) {
      table.pb(i);
      if (i * i != n)
        table.pb(n / i);
    }
    i++;
  }

  sort(all(table));

  return table;
}

ll char_to_idx(char c) {
  ll idx = 0;
  Forchar(cc, 'a', 'z') {
    if (c == cc)
      return idx;
    else
      idx++;
  }
}

ll next_combination(ll sub) {
  /*
  ### Attention ###
  if k is 0 or 1 and n is 0, it does not work well.

  ll n, k; ll bit = (1 << k) - 1;
  for (; bit < (1 << n); bit = next_combination(bit)){
      bool ith = bit & (1 << i);
      procedures...
  }

      sub & -sub: the binary which shares the last digit whose value is 1 in sub
      sub + x   : carry up the last digit
          ~y    : the binary whose digits are 1 if y's digit is 0.
  (sub & ~y) / x: reduce the same number of 0s after first 1 in x from (sub &
  ~y).
  */
  ll x = sub & -sub, y = sub + x;
  return (((sub & ~y) / x) >> 1) | y;
}

// just change the input if you want to change the target.
// If you want to check the common sequences in two strings,
// combine them. e.g. ABC150F
vl z_algorithm(str s) {
  /*
  Paramters
  ---------
  str: the string of interest

  Returns
  -------
  res[i] is the maximum number of K which satisfies
  s[:K] == s[i:i + K]
  for each i = 0, 1, 2, ..., n - 1.
  */
  ll n = s.length();
  vl res(n);
  res[0] = n;
  ll i1 = 1, i2 = 0;

  while (i1 < n) {
    /*
    i1: the starting point
    i2: the length of substring
    */
    while (i1 + i2 < n && s[i2] == s[i1 + i2])
      ++i2;
    res[i1] = i2;
    if (i2 == 0) {
      ++i1;
      continue;
    }
    ll i3 = 1;

    // update the already seen points
    while (i1 + i3 < n && i3 + res[i3] < i2) {
      res[i1 + i3] = res[i3];
      ++i3;
    }
    // update up to i1 + i3 and the next possible minimum length is i2 - i3 (=
    // res[i3])
    i1 += i3, i2 -= i3;
  }
  return res;
}

ll big_number_mod(str s, ll mod) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth) % mod;
    val %= mod;
    tenth *= 10;
    tenth %= mod;
    idx++;
  }

  return val;
}

ll big_number_compare(str s1, str s2) {
  if (s1.length() > s2.length())
    return 1;
  else if (s1.length() < s2.length())
    return -1;
  else if (s1 == s2)
    return 0;

  return 2 * (s1 > s2) - 1;
}

ll string_to_ll(str s) {
  ll l = s.length();
  ll idx = 0;
  ll val = 0;
  ll tenth = 1;
  while (idx < l) {
    ll m = ctol(s[l - 1 - idx]);
    val += (m * tenth);
    tenth *= 10;
    idx++;
  }

  return val;
}

str reflected_string(str s) {
  str t, u;
  ll n = s.length();
  t = s;
  reverse(all(t));
  u = substring(t, 0, n - 2) + s + substring(t, 1, n - 1);
  return u;
}

ld distance_between_point_line(point l_begin, point l_end, point p) {
  ll xl1 = l_begin.x, yl1 = l_begin.y;
  ll xl2 = l_end.x, yl2 = l_end.y;
  ll xp = p.x, yp = p.y;
  ll a = yl2 - yl1;
  ll b = -xl2 + xl1;
  ll c = -a * xl2 - b * yl2;
  return abs(ld(a * xp + b * yp + c)) / ld(sqrt(a * a + b * b));
}

bool is_cross(point l1_begin, point l1_end, point l2_begin, point l2_end) {
  ll x1 = l1_begin.x, y1 = l1_begin.y;
  ll x2 = l1_end.x, y2 = l1_end.y;
  ll x3 = l2_begin.x, y3 = l2_begin.y;
  ll x4 = l2_end.x, y4 = l2_end.y;
  ll val1 = (x1 - x2) * (y3 - y1) + (y1 - y2) * (x1 - x3);
  ll val2 = (x1 - x2) * (y4 - y1) + (y1 - y2) * (x1 - x4);
  ll val3 = (x3 - x4) * (y1 - y3) + (y3 - y4) * (x3 - x1);
  ll val4 = (x3 - x4) * (y2 - y3) + (y3 - y4) * (x3 - x2);

  return val1 * val2 < 0 && val3 * val4 < 0;
}

ld space_of_triangle(point p1, point p2, point p3) {
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;
  ll v1 = x2 - x1;
  ll u1 = y2 - y1;
  ll v2 = x3 - x1;
  ll u2 = y3 - y1;
  ld s = ld(v1 * u2 - u1 * v2) / ld(2);
  return abs(s);
}

pair<point, ll> center_2det_of_3points(point p1, point p2, point p3) {
  // the center of circle on the given three points
  // return the determinant value and the product of center points and 2 *
  // determinant value
  ll x1 = p1.x, y1 = p1.y;
  ll x2 = p2.x, y2 = p2.y;
  ll x3 = p3.x, y3 = p3.y;

  // center of 2 points
  ll c2x_2 = x2 + x1, c2y_2 = y2 + y1;
  ll c3x_2 = x3 + x1, c3y_2 = y3 + y1;

  // vertical vector of 2 lines
  ll b2y = x2 - x1, b2x = -y2 + y1;
  ll b3y = x3 - x1, b3x = -y3 + y1;

  ll x1_2 = c3x_2 * b3y - c3y_2 * b3x, y1_2 = c2x_2 * b2y - c2y_2 * b2x;
  ll det = -b3y * b2x + b3x * b2y;
  if (det == 0)
    return {{INF, INF}, det};
  ll cx_2det = -b2x * x1_2 + b3x * y1_2, cy_2det = -b2y * x1_2 + b3y * y1_2;
  return {{cx_2det, cy_2det}, det};
}

ll inversion_number(vl a, ll a_max) {
  /*
  Paramters
  ---------
  a: vector<ll>
      All the elements must be non-negative.
      Prefably the elements are compressed to reduce the computational cost.
  a_max: ll
      The maximum value of the vector a or the value bigger than the value
  stated previously.
  */

  BIT bit(a_max + 1);
  ll val = 0;
  rep(i, a.size()) {
    // i is the number of elements that have lower index than a[i].
    // call the number of elements that have lower value than a[i]
    // by subtracting these two, the residual number is the number of elements
    // that have larger value
    val += i - bit.query(a[i] - 1); // cumulative sum from 0 to a[i] - 1
    bit.update(a[i], 1);
  }
  return val;
}

template <typename T> vector<T> compress(vector<T> v) {
  // sort and remove all the duplicated values
  sort(all(v));
  v.erase(unique(all(v)), v.end());
  return v;
}

template <typename T> map<T, ll> dict(const vector<T> &v) {
  map<T, ll> d;
  rep(i, v.size()) d[v[i]] = i;
  return d;
}

points compress2D(vl xs, vl ys) {
  /*
  NOTE
  ----
  Add the corner points if required
  */

  ll n = xs.size();
  vl xcs = compress(xs), ycs = compress(ys);
  map<ll, ll> xd = dict(xcs), yd = dict(ycs);
  points ps(n);

  rep(i, n) xs[i] = xd[xs[i]], ys[i] = yd[ys[i]];

  rep(i, n) ps[i] = {xs[i], ys[i]};

  sort(all(ps));

  return ps;
}

void GaussJordanBitVector(vl &bs) {
  ll n = bs.size();
  ll rank = 0;
  ll j = 0;
  revrep(i, N_DIGITS) {
    for (j = rank; j < n; j++)
      if (bs[j] & (1LL << i))
        break;
    if (j == n)
      continue;
    if (j > rank)
      bs[rank] ^= bs[j];
    for (j = rank + 1; j < n; j++)
      bs[j] = min(bs[j], bs[j] ^ bs[rank]);
    rank++;
  }
}

ll kruskal(vector<undirected_edge> &es, ll n_vertex) {
  sort(all(es));
  UnionFind uf(n_vertex);

  ll min_cost = 0;

  rep(i, es.size()) {
    undirected_edge &e = es[i];
    if (!uf.is_same(e.from, e.to)) {
      min_cost += e.cost;
      uf.unite(e.from, e.to);
    }
  }
  return min_cost;
}

ll LongestIncreasedSequence(vl &v, ll n) {
  vl dp(n, INF);
  rep(i, n) * lower_bound(all(dp), v[i]) = v[i];
  return find(all(dp), INF) - dp.begin();
}

/*

diameter of tree
Graph tree[MAX_N];
ll dM = 0, vM = 0, v2 = 0;

void dfs1(ll source, ll parent, ll d){
    if (d > dM) dM = d, vM = source;

    rep(i, tree[source].size()){
        ll target = tree[source][i].to;
        if (target == parent) continue;
        dfs1(target, source, d + 1);
    }
}

void dfs2(ll source, ll parent, ll d){
    if (dM <= d) dM = d, v2 = source;
    rep(i, tree[source].size()){
        ll target = tree[source][i].to;
        if (target == parent) continue;
        dfs2(target, source, d + 1);
    }
}

dfs(0, -1, 0);
dfs2(vM, -1, 0);
prl2(vM + 1, v2 + 1); // the two edges of tree


const ll N_VERTEX = 310;
ll a, b, t;
vector<vl> dist;

void warshall_floyd(ll n){
    // rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(k, n) rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][k] +
dist[k][j]);
}

int main(void){
    in2(n, m);
    rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        dist[a][b] = t;
        dist[b][a] = t;
    }

    warshall_floyd(n);
}

vl dist, vertex_pre;

void dijkstra(ll start, ll n) {
    priority_queue<pl, vector<pl>, greater<pl>> edge_costs;
    dist = vl(n, INF); // vertex_pre = vl(n, -1);
    dist[start] = 0;
    edge_costs.push(pl(0, start));

    while (!edge_costs.empty()) {
        pl edge_cost = edge_costs.top();
        edge_costs.pop();
        ll idx = edge_cost.second;
        ll cost = edge_cost.first;
        if (dist[idx] < cost) continue;

        rep(i, graph[idx].size()){
            edge e = graph[idx][i];
            if (dist[e.to] > dist[idx] + e.cost){
                dist[e.to] = dist[idx] + e.cost;
                // vertex_pre[e.to] = idx;
                edge_costs.push(pl(dist[e.to], e.to));
            }
        }
    }
}

vl get_predecessor(ll g){
    vl path;
    for (; g != -1; g = vertex_pre[g]) path.pb(g);

    reverse(all(path));
    return path;
}

int main(void){
    in2(n, m);
    rep(i, m){
        in3(a, b, t);
        a--; b--;
        G[a].pb({b, t});
        G[b].pb({a, t});
    }

    dijkstra(0, n);
}


# ABC061D

bool find_negative_cycle(ll goal){
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[e.to] != INF && dist[e.to] > dist[v] + e.cost){
            dist[e.to] = -INF;
            if (goal == -1) return true;
            else if (goal == e.to) return true;
        }
    }
    return false;
}

bool bellman_ford(ll start, ll n, ll goal){
    // if there is a closed circuit, it returns false. (when goal == -1)
    // if the distance to goal cannot be obtained, it returns false (when goal
!= -1) fill(dist, dist + n, INF); dist[start] = 0; rep(i, n) rep(v, n) rep(k,
graph[v].size()){ edge e = graph[v][k]; if (dist[v] != INF && dist[e.to] >
dist[v] + e.cost) dist[e.to] = dist[v] + e.cost;
    }

    if (find_negative_cycle(goal)) return false;
    return true;
}

*/

/*
    # 1. The usage of map pair
    map<pl, ll> cnt;
    cnt[{i, j}] = 0;
    items(kv, cnt){
        prl2(kv.first, kv.second);
    }

    # 2. The usage of next_permutation and combination (factorial search)
    ll a[8];
    rep(i, 8) a[i] = i;
    sort(a, a + 8);
    do{
    }while(next_permutation(a, a+n));

    // here, combination
    ll n, r;
    ll i1, i2, ..., ir;
    For(i1, n - r, n) For(i2, n - r - 1, i1) ... For(ir, n - 2 * r + 1, irr){
   process;}

    # 3. bit search
    ll n;
    in1(n);
    const ll base = 3;
    ll upper = power_normal(base, n);

    rep(i, upper){
        ll tmp = i;
        rep(j, n){
            rep(k, base) if (tmp % base == k) prl(k);
            tmp /= base;
        }
    }

    # 4. imos method
    // used when we would like to count the number which
    // shows how many times the numbers between l and r belongs to smt.
    // This method is composed of three process.
    ll n, m, s[MAX_M], l, r;
    in2(n, m);
    rep(i, m) s[i] = 0;

    // 1st step
    rep(i, n){
        in3(l, r, c);
        l--; r--; // if l starts from 1.
        s[l] += c; s[r + 1] -= c;
    }

    // 2nd step
    rep(i, m - 1) s[i + 1] += s[i];
    // 3rd step: judgement...

    #5. shakutori method (syakutori, two pointers technique)
    // 1. strech right side while the condition is met.
    // 2. renew the answer
    // 3. increments left side
    // 4. Back to 1. (l <= r must be satisfied all the time.)

    ll l = 0; ll r = 0;
    while (l < n){
        r = max(r, l);
        if (l == r) r++;
        while(r < n && cond) r++;
        answer += r - l; l++;
    }
    prl(answer);

    #6. priority queue

    pq q;
    ll answer = 0;
    ll v;

    rep(i, n) q.push(a[i]);
    rep(i, m){
        v = q.top(); q.pop(); // get the top value and dump the value from queue
        v /= 2; q.push(v); // add the new value
    }

    while(!q.empty()){
        answer += q.top();
        q.pop();
    }

    #7. The shortest path (distance) between the k-th edge and another edge
   (Tree) Graph tree[MAX_N]; ll depth[MAX_N];

    void dfs(ll source, ll parent, ll all_cost){
        depth[source] = all_cost;
        items(e, tree[source]){
            if (e.to == parent) continue;
            dfs(e.to, source, all_cost + e.cost);
        }
    }

    ll n, k, a, b, c;

    in2(n, k);
    rep(i, n - 1){
        in3(a, b, c);
        a--; b--;
        tree[a].pb({b, c});
        tree[b].pb({a, c});
    }

    k--;
    dfs(k, -1, 0);

    #10. Visiting Subtree using recurrent function (ABC138D)

    Graph tree[MAX_N];
    ll c[MAX_N];
    bool visited[MAX_N];

    void dfs(ll source, ll parent, ll val){
        visited[source] = true;
        c[source] += val;
        rep(i, tree[source].size()){
            rep(i, m){ll res = n % match[i].e1;}
            ll vertex = tree[source][i].to;
            if (vertex == parent) continue;
            dfs(vertex, source, c[source]);
        }
    }

    #11. bfs ABC146D, ABC007C
    1. first create a tree.
    2. start searching from a node.
    3. do some processes and push nodes connected with a given target node in
   BFS.
    4. repeat a series of procedure until queue is empty.

    queue<pl> q;

    void bfs(ll source, ll parents){
        ll n_edge = graph[source].size();
        if (parents != -1) dist[source] = min(dist[source], dist[parents] + 1);
        if (visited[source]) return;
        visited[source] = true;

        rep(idx, n_edge){
            ll target = graph[source][idx].to;
            if (target == parents) continue;
            q.push(mp(target, source));
        }
    }

    q.push(mp(sg.e1, -1));
    while(!q.empty()){
        pl source = q.front(); q.pop();
        bfs(source.e1, source.e2);
    }

    #12. grid to distance matrix (dx, dy)
    ll w, h;
    ll pos_to_idx(ll x, ll y){
        return y * w + x;
    }

    pl idx_to_pos(ll idx){
        return mp(idx % w, idx / w);
    }

    rep(y, h){
        in1(s);
        rep(x, w){
            if (s[x] == '#') wall[x][y] = true;
            else wall[x][y] = false;
        }
    }

    rep(i1, h * w)rep(i2, h * w) dist[i1][i2] = INF * (i1 != i2);

    rep(x, w)rep(y, h){
        ll idx1 = pos_to_idx(x, y); ll idx2;
        if (wall[x][y]) continue;
        if (x != 0 && !wall[x - 1][y]){
            idx2 = pos_to_idx(x - 1, y);
            // if warshall floyd
            dist[idx1][idx2] = 1;
            // if dijkstra
            // graph[idx1].pb({idx2, 1});
        }
        if (x != w - 1 && !wall[x + 1][y]){
            idx2 = pos_to_idx(x + 1, y);
            dist[idx1][idx2] = 1;
            // graph[idx1].pb({idx2, 1});
        }
        if (y != 0 && !wall[x][y - 1]){
            idx2 = pos_to_idx(x, y - 1);
            dist[idx1][idx2] = 1;
            // graph[idx1].pb({idx2, 1});
        }
        if (y != h - 1 && !wall[x][y + 1]){
            idx2 = pos_to_idx(x, y + 1);
            dist[idx1][idx2] = 1;
            // graph[idx1].pb({idx2, 1});
        }
    }

    # Cumulative sum (2 dimension)

    ll func(ll x, ll y, ll dx, ll dy){
        if (x + dx > w || y + dy > h) return - INF;
        ll val = cum[x + dx][y + dy];
        val += cum[x][y];
        val -= cum[x][y + dy];
        val -= cum[x + dx][y];
        return val;
    }

    rep(x, w + 1) cum[x][0] = 0;
    rep(y, h + 1) cum[0][y] = 0;
    rep(y, h + 1) rep(x, w)
        cum[x + 1][y + 1] = cum[x][y + 1] + vs[x][y];

    rep(x, w + 1) rep(y, h)
        cum[x][y + 1] += cum[x][y];

    # DEBUG_MATRIX

    rep(i, n){
        rep(j, n) prs(a[i][j]);
        prl("");
    }

*/

/*
# the operators regarding bit
& (AND), | (OR), ^ (XOR)
- (REVERSE), >> (SMALLER SHIFT)
<< (BIGGER SHIFT)


     x1: 0000 0001 0010 0101 0110 0111 0111
     x2: xxxx 0001 0011 0100 0101 1000 0110
x1 & x2: 0000 0001 0010 0100 0100 0000 0110

      x: 1001 1010  1100 1011 1101  1111
x & - x: 0001 0010  0100 0001 0001  0001
    sum: 1010 1100 10000 1100 1100 10000

x << y is x * 2 ** y
x >> y is rep(i, y) x = x // 2

####### Attention #######
1 << i and 1LL << i is different. If programs show WA, try switch to this one.
Let S be a bit sequence and i be a non-negative integer
S & (1 << i) -> if true, i in S
S | (1 << i) -> S union {i}
S & -(1 << i) -> S - {i}
__builtin_popcount(S) -> the number of elements in S
S = 0 -> S is an empty set
__builtin_popcountl(i) -> the number of 1 in binary
S = (1 << n) - 1 -> S includes all the elements up to the n-th

#Conditional Operator
condition ? true : false;

#iterator
type declaration: auto
value reference: *itr
increment: itr++
decrement: itr--
substitution of value: *itr = smt

# inclusion-exclusion principle
|U[i = 1 to n] Ai| = sum[i = 1 to n] |Ai| - sum[i < j]|Ai ^ Aj| + ... + (-1)^(n
- 1) |^[i = 1 to n]Ai|
*/

const ll MAX_N = 200005;
bool okay = false;
ll answer = 0;
ll n, m, k;

// ABC160F is one example
// Modify the functions evaluate and merge based on given problems
struct Merger {
  using type = ll;

  static type merge(const vector<type> &value) {
    // merge the result below the source node
    // each value is from each child node of the source node
    // value[i] := f(child i)
    // Here, we would like to obtain f(source) using f(child i) (i = 0, 1, ...,
    // n_children)
    ll ret = 1;
    for (auto &&v : value)
      ret += v;
    return ret;
  }

  static vector<type> evaluate(const vector<type> &value) {
    // value[i] := f(source -> child i)
    // we would like to obtain f(child i -> source)
    // child j (j != i) is the grandchildren of child i
    // represent f(child i -> source) using f(source -> j) (j != i)
    // L[i + 1] := the result using f(source -> k) (k = 0, 1, ..., i)
    // R[i] := the result using f(source -> k) (k = i, i + 1, ..., n_children)

    const ll n_children = value.size();
    vl L(n_children + 1, 0), R(n_children + 1, 0);
    rep(i, n_children) L[i + 1] = L[i] + value[i];
    revrep(i, n_children) R[i] = R[i + 1] + value[i];

    vl ret(n_children);
    rep(i, n_children) ret[i] = L[i] + R[i + 1] + 1;

    return ret;
  }
};

ll memo[MAX_N];

void solve() {
  RerootingTreeDP<Merger> dp(n);
  rep(i, n - 1) {
    ll a, b;
    in2(a, b);
    a--, b--;
    dp.add_edge(a, b, 1);
  }

  memo[0] = 1;
  rep(i, n + 3) memo[i + 1] = (memo[i] * 2) % MOD;
  dp.dp();
  rep(i, n) { // dp.tree[i] := the number of children
    (answer += memo[n - 1] - 1) %=
        MOD;                    // all the cases except all of them are white
    for (auto &&e : dp.tree[i]) // the case where the i-th vertex is not a hole
      (answer += MOD - (memo[e.value] - 1)) %= MOD;
  }

  ll x_inv = modinv(power_mod(2, n, MOD), MOD);
  ll y = answer;

  prl((y * x_inv) % MOD);
}

int main(void) {
  in1(n);
  solve();
  return 0;
}