Problem p02686 (C++) — buggy submission, verdict: Runtime Error

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
// const ll MOD = 1000000007;
const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-12;

struct mint {
  ll x;
  mint(ll x = 0) : x((x % MOD + MOD) % MOD) {}
  mint operator-() const { return mint(-x); }

  mint &operator+=(const mint a) {
    if ((x += a.x) >= MOD)
      x -= MOD;
    return *this;
  }

  mint &operator-=(const mint a) {
    if ((x += MOD - a.x) >= MOD)
      x -= MOD;
    return *this;
  }

  mint &operator*=(const mint a) {
    (x *= a.x) %= MOD;
    return *this;
  }
  mint operator+(const mint a) const { return mint(*this) += a; }
  mint operator-(const mint a) const { return mint(*this) -= a; }
  mint operator*(const mint a) const { return mint(*this) *= a; }
};
istream &operator>>(istream &is, const mint &a) { return is >> a.x; }
ostream &operator<<(ostream &os, const mint &a) { return os << a.x; }

using edge = struct {
  ll to;
  ll cost;
};

struct fpoint {
  ld x;
  ld y;
  bool operator<(const fpoint &p) const {
    if (x == p.x)
      return y < p.y;
    return x < p.x;
  }

  bool operator==(const fpoint &p) const {
    if (p.x - EPS < x && x < p.x + EPS && p.y - EPS < y && y < p.y + EPS)
      return true;
    return false;
  }

  bool operator!=(const fpoint &p) const {
    if (p.x - EPS > x || x > p.x + EPS || p.y - EPS > y || y > p.y + EPS)
      return true;
    return false;
  }

  fpoint &operator+=(const ld a) {
    x += a, y += a;
    return *this;
  }

  fpoint &operator-=(const ld a) {
    x -= a, y -= a;
    return *this;
  }

  fpoint &operator*=(const ld a) {
    x *= a, y *= a;
    return *this;
  }

  fpoint &operator/=(const ld a) {
    x /= a, y /= a;
    return *this;
  }

  fpoint &operator+=(const fpoint p) {
    x += p.x, y += p.y;
    return *this;
  }

  fpoint &operator-=(const fpoint p) {
    x -= p.x, y -= p.y;
    return *this;
  }

  fpoint &operator*=(const fpoint p) {
    x *= p.x, y *= p.y;
    return *this;
  }

  fpoint &operator/=(const fpoint p) {
    x /= p.x, y /= p.y;
    return *this;
  }

  bool is_zero() {
    fpoint z = {0, 0};
    return z == *this;
  }

  fpoint operator+(const fpoint p) const { return fpoint(*this) += p; }
  fpoint operator-(const fpoint p) const { return fpoint(*this) -= p; }
  fpoint operator*(const fpoint p) const { return fpoint(*this) *= p; }
  fpoint operator/(const fpoint p) const { return fpoint(*this) /= p; }
  fpoint operator+(const ld a) const { return fpoint(*this) += a; }
  fpoint operator-(const ld a) const { return fpoint(*this) -= a; }
  fpoint operator*(const ld a) const { return fpoint(*this) *= a; }
  fpoint operator/(const ld a) const { return fpoint(*this) /= a; }
};

struct point {
  ll x;
  ll y;
  bool operator<(const point &p) const {
    if (x == p.x)
      return y < p.y;
    return x < p.x;
  }

  bool operator==(const point &p) const {
    if (x == p.x && y == p.y)
      return true;
    return false;
  }

  bool operator!=(const point &p) const {
    if (x != p.x || y != p.y)
      return true;
    return false;
  }

  point &operator+=(const ll a) {
    x += a, y += a;
    return *this;
  }

  point &operator-=(const ll a) {
    x -= a, y -= a;
    return *this;
  }

  point &operator*=(const ll a) {
    x *= a, y *= a;
    return *this;
  }

  point &operator+=(const point p) {
    x += p.x, y += p.y;
    return *this;
  }

  point &operator-=(const point p) {
    x -= p.x, y -= p.y;
    return *this;
  }

  point &operator*=(const point p) {
    x *= p.x, y *= p.y;
    return *this;
  }

  bool is_zero() {
    point z = {0, 0};
    return z == *this;
  }

  fpoint to_fpoint() {
    fpoint ret = {ld(x), ld(y)};
    return ret;
  }

  void operator++(int) { x++, y++; }
  void operator++() { x++, y++; }
  void operator--(int) { x--, y--; }
  void operator--() { x--, y--; }
  point operator+(const point p) const { return point(*this) += p; }
  point operator-(const point p) const { return point(*this) -= p; }
  point operator*(const point p) const { return point(*this) *= p; }
  point operator+(const ll a) const { return point(*this) += a; }
  point operator-(const ll a) const { return point(*this) -= a; }
  point operator*(const ll a) const { return point(*this) *= a; }
};

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::tuple<ll, ll, ll> tp3;
typedef std::tuple<ll, ll, ll, ll> tp4;
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<vl> vl2;
typedef std::vector<vl2> vl3;
typedef std::vector<vl3> vl4;
typedef std::vector<mint> vmi;
typedef std::vector<vmi> vmi2;
typedef std::vector<vmi2> vmi3;
typedef std::vector<vmi3> vmi4;
typedef std::vector<bool> vb;
typedef std::vector<vb> vb2;
typedef std::vector<vb2> vb3;
typedef std::vector<vb3> vb4;
typedef std::vector<pl> vpl;
typedef std::vector<tp3> vtp3;
typedef std::vector<tp4> vtp4;
typedef std::vector<point> points;
typedef std::vector<fpoint> fpoints;
// priority queue. Taking from the higher value. Don't forget calling !q.empty()
typedef std::priority_queue<ll> pq;
// priority queue. Taking from the lower value
typedef std::priority_queue<ll, vl, greater<ll>> pql;
typedef std::vector<vector<edge>> Graph;
const ll N_DIGITS = 60;
const long double PI = 3.14159265358979323846;
points 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(mint a) { return to_string(a.x); }

string to_string(point p) {
  return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";
}

string to_string(fpoint p) {
  return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";
}

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 mt make_tuple
#define pr(a) std::cout << (a)
#define prl(a) std::cout << (a) << endl
#define prl2(a, b) std::cout << (a) << " " << (b) << endl
#define prl3(a, b, c) std::cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a, b, c, d)                                                       \\
  std::cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) std::cout << (a) << " "
#define prs2(a, b) std::cout << (a) << " " << (b) << " "
#define prs3(a, b, c) std::cout << (a) << " " << (b) << " " << (c) << " "
#define prs4(a, b, c, d)                                                       \\
  std::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 in5(a, b, c, d, e) cin >> (a) >> (b) >> (c) >> (d) >> (e)
#define in6(a, b, c, d, e, f) cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f)
#define in7(a, b, c, d, e, f, g)                                               \\
  cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f) >> (g)
#define e1 first
#define e2 second
#define ltos(n) to_string((n))
#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)                                                    \\
  std::cout << fixed << setprecision(digits) << (num) << endl;
#define prl_time(s)                                                            \\
  prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))

#ifdef _LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif

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) {
    // complexity O(EV^2)
    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;
    }
  }
};

struct UnionFind {
  vl parents, set_size;
  set<ll> root_idx;
  ll n_groups;

  UnionFind(ll n) {
    parents = set_size = vl(n);
    n_groups = n;
    rep(i, n) {
      parents[i] = i;
      set_size[i] = 1LL;
      root_idx.insert(i);
    }
  }

  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];
    root_idx.erase(y);
    n_groups--;
  }

  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)]; }

  ll num_union() const { return n_groups; }
};

struct Doubling { // ABC167D
  ll n;
  ll sz;
  vl2 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 LowestCommonAncestor {
public:
  ll N, logN;
  vl depth, len;
  Graph tree;
  vl2 parents;

  LowestCommonAncestor(ll n, ll offset = -1, bool is_weighted = true) {
    N = n;
    logN = 0;
    while (N > (1LL << logN))
      logN++;
    depth = len = vl(N);
    parents = vl2(logN, vl(N));
    tree = Graph(N);
    input(N, offset, is_weighted);
    init(0, -1, 0, 0);
    build();
  }

  void add_edge(ll from, ll to, ll dist) {
    tree[from].pb({to, dist});
    tree[to].pb({from, dist});
  }

  void input(ll n, ll offset = -1, bool is_weighted = true) {
    rep(i, n - 1) {
      ll a, b, d = 1;
      in2(a, b);
      a += offset, b += offset;
      if (is_weighted)
        in1(d);
      add_edge(a, b, d);
    }
  }

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

  //       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, d);
    // 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)
    */

    if (i < 0)
      return ini;
    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) {
    // a[l] + a[l + 1] + ... + a[r - 1] + a[r]
    return query(r) - query(l - 1);
  }

  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() {
    vb 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(); }
};

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

  // This is the exaple of the number of children
  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;
  }
};

struct StronglyConnectedComponents {
  ll n, n_cmp;
  vl2 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 n_cmp = num;
  }
};

struct CombinationMemo {
  ll sz, mod;
  vl facts, facts_inv, minv;

  CombinationMemo(ll sz, ll _mod) : sz(sz), mod(_mod) {
    facts.resize(sz + 5);
    facts_inv.resize(sz + 5);
    minv.resize(sz + 5);
    init();
  }

  void init() {
    facts[0] = facts[1] = 1;
    minv[1] = 1;
    facts_inv[0] = facts_inv[1] = 1;

    For(i, 2, sz + 3) {
      facts[i] = (i * facts[i - 1]) % mod;
      minv[i] = mod - minv[mod % i] * (mod / i) % mod;
      facts_inv[i] = facts_inv[i - 1] * minv[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;
  }

  ll nPk(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;
    return val % mod;
  }
};

struct PowerMemo {
  ll sz, mod, base;
  vl powB;

  PowerMemo(ll sz, ll base, ll _mod) : sz(sz), base(base), mod(_mod) {
    powB.resize(sz + 5);
    init();
  }

  void init() {
    powB[0] = 1;
    rep(i, sz + 3) powB[i + 1] = (powB[i] * base) % mod;
  }

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

struct Grid2D {
  Graph graph;
  ll Width, Height;

  Grid2D(ll w, ll h) : Width(w), Height(h) { graph.resize(w * h); }

  ll pos_to_idx(point p) { return p.y * Width + p.x; }

  point idx_to_pos(ll idx) { return {idx % Width, idx / Width}; }

  bool undefined_region(point p, vb2 &block) {
    if (p.x < 0 || p.x > Width - 1)
      return true;
    if (p.y < 0 || p.y > Height - 1)
      return true;
    if (block[p.x][p.y])
      return true;
    return false;
  }

  void build(vb2 &block, ll val = 1) {
    rep(x, Width) rep(y, Height) {
      point p = {x, y};
      ll idx1 = pos_to_idx(p);
      ll idx2;

      if (block[x][y])
        continue;
      rep(i, 4) {
        point nxt = p + dirs[i];
        idx2 = pos_to_idx(nxt);
        if (!undefined_region(nxt, block))
          graph[idx1].pb(
              {idx2, val}); // dist[idx1][idx2] = val; (warshall-floyd)
      }
    }
  }
};

struct Cumulative2D {
  vl2 cum;
  ll w, h;

  Cumulative2D(ll w, ll h) : w(w), h(h) { cum = vl2(w + 1, vl(h + 1, 0)); }

  template <typename T> void build(vector<T> &vec) {
    // never forget building
    rep(x, w + 1) cum[x][0] = 0;
    rep(y, h + 1) cum[0][y] = 0;
    rep(y, h) rep(x, w) cum[x + 1][y + 1] = cum[x][y + 1] + vec[x][y];
    rep(x, w + 1) rep(y, h) cum[x][y + 1] += cum[x][y];
  }

  ll func(ll x, ll y, ll dx, ll dy) {
    // 1-indexed
    // the rectangle of (x, y), (x + dx, y), (x, y + dy) and (x + dx, y + dy)
    // think about the case of (1, 1, 1, 1).
    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;
  }
};

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_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 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 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;
}

vl2 pascal_triangle(ll n) {
  /*
  Complexity: O(n^2)
  The upper bound of n is nearly 50.

  Parameters
  ----------
  n; the size of returned vector

  Returns
  -------
  comb[i][j]: combination(i, j). 0 <= i <= n, 0 <= j <= i
  */

  vl2 comb(n + 1, vl(n + 1));
  comb[0][0] = 1;
  For(i, 1, n + 1) rep(j, i + 1) {
    comb[i][j] += comb[i - 1][j];
    if (j > 0)
      comb[i][j] += comb[i - 1][j - 1];
  }

  return comb;
}

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;
}

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

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;
}

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 next_combination(ll sub) {
  /*
  nCk
  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;

  if (x != 0)
    return (((sub & ~y) / x) >> 1) | y;
  else
    return INF;
}

// 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
template <typename T> vl z_algorithm(T &s, ll n) {
  /*
  Paramters
  ---------
  T: the string or list of interest
  n: the size of string or list

  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.
  */
  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 = s[l - 1 - idx] - '0';
    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 = s[l - 1 - idx] - '0';
    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;
}

template <typename T> bool isColinear(T p1, T p2, T p3) {
  T v1 = p2 - p1, v2 = p3 - p1;
  return v1.x * v2.y == v1.y * v2.x;
}

template <typename T> T PerpendicularBisector(T p1, T p2) {
  T vec = p2 - p1;
  assert(!vec.is_zero());
  T ret = {vec.y, -vec.x};

  return ret;
}

template <typename T> ld Distance2DPoints(T p1, T p2) {
  T vec = (p1 - p2) * (p1 - p2);
  return sqrt(vec.x + vec.y);
}

ll SquaredDistance2DPoints(point p1, point p2) {
  point vec = (p1 - p2) * (p1 - p2);
  return vec.x + vec.y;
}

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> OuterCenter(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
  point ret;

  if (isColinear(p1, p2, p3)) {
    ll d1 = SquaredDistance2DPoints(p1, p2);
    ll d2 = SquaredDistance2DPoints(p2, p3);
    ll d3 = SquaredDistance2DPoints(p3, p1);
    if (d1 >= d2 && d1 >= d3) {
      ret = p1 + p2;
      return mp(ret, 2);
    } else if (d2 >= d1 && d2 >= d3) {
      ret = p2 + p3;
      return mp(ret, 2);
    } else {
      ret = p3 + p1;
      return mp(ret, 2);
    }
  }

  point pv1 = PerpendicularBisector(p1, p2);
  point pv2 = PerpendicularBisector(p1, p3);
  point cv1_2x = p1 + p2, cv2_2x = p1 + p3;
  // cv1 + k pv1 == cv2 + m pv2
  // (pv1x -pv2x) (k) = (cv2x - cv1x)
  // (pv1y -pv2y) (m) = (cv2y - cv1y)
  ll det_inv = -pv1.x * pv2.y + pv1.y * pv2.x;
  ll x1_2x = cv2_2x.x - cv1_2x.x, x2_2x = cv2_2x.y - cv1_2x.y;
  pl c_2x_det = {-pv2.y * x1_2x + pv2.x * x2_2x,
                 -pv1.y * x1_2x + pv1.x * x2_2x};

  // ret.x = ld(cv1_2x.x * det_inv + pv1.x * c_2x_det.e1) / ld(2 * det_inv);
  // ret.y = ld(cv1_2x.y * det_inv + pv1.y * c_2x_det.e1) / ld(2 * det_inv);
  ret.x = cv1_2x.x * det_inv + pv1.x * c_2x_det.e1;
  ret.y = cv1_2x.y * det_inv + pv1.y * c_2x_det.e1;

  ll jacobian = 2 * det_inv;
  return mp(ret, jacobian);
}

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.
  */

  BinaryIndexedTree 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;
}

ld bin_search(ld left, ld right, bool lb, function<bool(ld)> judge) {
  ld mid;
  while (right - left > EPS) {
    mid = (right + left) / 2;
    if (lb) {
      if (judge(mid))
        right = mid;
      else
        left = mid + EPS;
    } else {
      if (judge(mid))
        left = mid;
      else
        right = mid - EPS;
    }
  }
  return right;
}

ll bin_search(ll left, ll right, bool lb, function<bool(ll)> judge) {
  ll mid;
  while (right > left) {
    if (lb) {
      // if true (satisfies the condition), range shifts smaller direction
      mid = (right + left) / 2;
      if (judge(mid))
        right = mid;
      else
        left = mid + 1;
    } else {
      // if true (satisfies the condition), range shitfs larger direction
      mid = (right + left + 1) / 2;
      if (judge(mid))
        left = mid;
      else
        right = mid - 1;
    }
  }
  return right;
}

ld trinary_search(ld left, ld right, function<ld(ld)> func) {
  // Care the value EPS!!! Compare to the condition
  while (abs(right - left) > EPS) {
    ld left2 = (2.0 * left + right) / 3.0;
    ld right2 = (left + 2.0 * right) / 3.0;
    ld f1 = func(left2);
    ld f2 = func(right2);
    if (f1 <= f2)
      right = right2;
    else if (f2 <= f1)
      left = left2;
  }

  return right;
}

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) {
  // O(ElogE)
  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 = v.size();
  vl dp(n, INF);
  rep(i, n) * lower_bound(all(dp), v[i]) = v[i];
  return lower_bound(all(dp), INF) - dp.begin();
}

void dijkstra(ll start, Graph &graph, vl &dist, vl &vertex_pre,
              bool trace = false) {
  priority_queue<pl, vpl, greater<pl>> edge_costs;
  ll n = graph.size();
  dist = vl(n, INF);
  if (trace)
    vertex_pre = vl(n, -1);
  dist[start] = 0;
  edge_costs.push(pl(0, start));

  while (!edge_costs.empty()) {
    ll idx, cost;
    tie(cost, idx) = edge_costs.top();
    edge_costs.pop();
    if (dist[idx] < cost)
      continue;

    for (auto e : graph[idx]) {
      if (dist[e.to] > dist[idx] + e.cost) {
        dist[e.to] = dist[idx] + e.cost;
        if (trace)
          vertex_pre[e.to] = idx;
        edge_costs.push(pl(dist[e.to], e.to));
      }
    }
  }
}

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

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

void warshall_floyd(vl2 &dist) {
  ll n = dist.size();
  // Dont forget the initialization
  // 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]);
}

// ABC061D
bool find_negative_cycle(ll n, ll goal, Graph &graph, vl &dist) {
  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 goal, Graph &graph, vl &dist) {
  // 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)
  ll n = graph.size();
  dist = vl(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(n, goal, graph, dist))
    return false;
  return true;
}

void Euler_Tour(Graph &tree, vl &euler_tour, vl &in, vl &out, ll anc = 0) {
  // record only when we first reach a node and leave the node.
  ll n = tree.size();
  vb visited(n, false);
  in = out = vl(n);

  auto dfs = [&](auto dfs, ll source) -> void {
    visited[source] = true;
    in[source] = euler_tour.size();
    euler_tour.pb(source);

    bool is_leaf = true;
    for (auto &e : tree[source]) {
      ll target = e.to;
      if (visited[target])
        continue;
      else
        is_leaf = false;
      dfs(dfs, target);
    }

    euler_tour.pb(-source);
    out[source] = euler_tour.size() - 1;
  };

  dfs(dfs, anc);
}

void Euler_Tour2(Graph &tree, vl &euler_tour, vl &in, vl &out, ll anc = 0) {
  // record everytime we reach a node
  ll n = tree.size();
  vb visited(n, false);
  in = out = vl(n);

  auto dfs = [&](auto dfs, ll source) -> void {
    visited[source] = true;
    in[source] = euler_tour.size();
    euler_tour.pb(source);

    bool is_leaf = true;
    for (auto &e : tree[source]) {
      ll target = e.to;
      if (visited[target])
        continue;
      else
        is_leaf = false;
      dfs(dfs, target);
      euler_tour.pb(source);
    }

    out[source] = euler_tour.size() - 1;
  };

  dfs(dfs, anc);
}

std::mt19937 create_rand_engine() {
  std::random_device rnd;
  std::vector<std::uint_least32_t> v(10);
  std::generate(v.begin(), v.end(), std::ref(rnd));
  std::seed_seq seed(v.begin(), v.end());
  return std::mt19937(seed);
}

vl generate_unique_array(const size_t sz, ll vm, ll vM) {
  const size_t range = static_cast<size_t>(vM - vm + 1);
  const size_t num = static_cast<size_t>(sz * 1.2);
  assert(vm <= vM);
  assert(sz <= range);

  vl ret;
  auto engine = create_rand_engine();
  std::uniform_int_distribution<ll> distribution(vm, vM);

  while (ret.size() < sz) {
    while (ret.size() < num)
      ret.pb(distribution(engine));
    sort(all(ret));

    auto unique_end = unique(all(ret));
    if (sz < distance(ret.begin(), unique_end))
      unique_end = next(ret.begin(), sz);

    ret.erase(unique_end, ret.end());
  }

  return ret;
}

vl generate_array(const size_t sz, ll vm, ll vM) {
  const size_t range = static_cast<size_t>(vM - vm + 1);
  assert(vm <= vM);

  vl ret;
  auto engine = create_rand_engine();
  std::uniform_int_distribution<ll> distribution(vm, vM);

  while (ret.size() < sz)
    ret.pb(distribution(engine));

  return ret;
}

void read_graph(ll n_vertex, ll n_edges, Graph &graph, ll cost = -1,
                bool directed = false, ll offset = -1) {

  graph.resize(n_vertex);
  rep(i, n_edges) {
    ll from, to, c = cost;
    if (cost == -1)
      in3(from, to, c);
    else
      in2(from, to);
    from += offset, to += offset;

    graph[from].pb({to, cost});
    if (!directed)
      graph[to].pb({from, cost});
  }
}

void read_vector(ll n, vl &v, ll offset = 0) {
  v.resize(n);
  rep(i, n) {
    in1(v[i]);
    v[i] += offset;
  }
}

void read_pair_vector(ll n, vpl &v, ll offset1 = 0, ll offset2 = 0,
                      bool rev = false) {
  v.resize(n);
  rep(i, n) {
    ll a, b;
    in2(a, b);
    a += offset1, b += offset2;
    if (!rev)
      v[i] = {a, b};
    else
      v[i] = {b, a};
  }
}

void read_2DMap(ll w, ll h, vb2 &block, char b) {
  block = vb2(w, vb(h, false));
  str s;
  rep(y, h) {
    in1(s);
    rep(x, w) block[x][y] = (s[x] == b);
  }
}

/*

diameter of tree
Graph tree;
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

*/

/*

    # 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));

    # 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){
        if (l == r) r++;
        while(r < n && cond) r++;
        l++;
    }
    prl(answer);

    #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);
    }
*/

/*
     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

####### Attention #######
S & (1 << i) -> if true, i in S
S | (1 << i) -> S union {i}
S & ~(1 << i) -> S - {i}
__builtin_popcountl(i) -> the number of 1 in binary

#Conditional Operator
condition ? true : false;

# 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;
const size_t MAX_BIT = 100;
typedef bitset<MAX_BIT> bts;
typedef vector<bts> vbt;
typedef vector<vbt> vbt2;
typedef vector<vbt2> vbt3;
bool okay = false;
ll answer = 0;
str s, t, u;
ll n, m, h, w, k, Q;
ll a, b, c, d, idx, cnt;
vl as, bs, cs, ds;
vb2 block;
vb visited;
Graph graph;
Graph tree;
/*
read_graph(n, m, graph, cost, directed, offset)
    cost: if link cost is fixed, put positive integer
          otherwise -1
    directed: if directed true, olse false
    offset: if the given vertex is 1-indexed, -1.
*/

void solve() {
  // generate_array(size, min, max);
  in1(n);

  multiset<pl> rs, ls;
  rep(i, n) {
    in1(s);
    ll cur = 0, lb = 0;
    rep(i, s.length()) {
      if (s[i] == '(')
        cur++;
      else
        cur--;
      chmin(lb, cur);
    }
    if (cur >= 0)
      ls.insert({-lb, cur});
    else
      rs.insert({-(lb - cur), -cur});
  }

  ll cur = 0;
  auto it = ls.begin();
  while (it != ls.end()) {
    if (cur - (*it).e1 < 0) {
      prl("No");
      return;
    }
    cur += (*it).e2;
    it++;
  }

  ll target = cur;
  cur = 0;
  it = rs.begin();
  while (it != rs.end()) {
    if (cur - (*it).e1 < 0) {
      prl("No");
      return;
    }
    cur += (*it).e2;
    it++;
  }

  yn(target == cur);

  auto naive_solver = [&]() {};
#ifdef _LOCAL
  naive_solver();
#endif
}

void test(ll num = 0, bool verbose = false) {
  rep(i, max(1LL, num)) {
    ll t = clock();
    if (verbose)
      prl4("\\n#####", i + 1, "#####", "\\n## Answer ##");
    solve();
    if (verbose) {
      prl("");
      prl_time(t);
    }
  }
}

int main(void) {
#ifdef _LOCAL
  test(4, true);
#else
  test(0, false);
#endif

  return 0;
}


Error: 0

Fixed submission:

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
// const ll MOD = 1000000007;
const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-12;

struct mint {
  ll x;
  mint(ll x = 0) : x((x % MOD + MOD) % MOD) {}
  mint operator-() const { return mint(-x); }

  mint &operator+=(const mint a) {
    if ((x += a.x) >= MOD)
      x -= MOD;
    return *this;
  }

  mint &operator-=(const mint a) {
    if ((x += MOD - a.x) >= MOD)
      x -= MOD;
    return *this;
  }

  mint &operator*=(const mint a) {
    (x *= a.x) %= MOD;
    return *this;
  }
  mint operator+(const mint a) const { return mint(*this) += a; }
  mint operator-(const mint a) const { return mint(*this) -= a; }
  mint operator*(const mint a) const { return mint(*this) *= a; }
};
istream &operator>>(istream &is, const mint &a) { return is >> a.x; }
ostream &operator<<(ostream &os, const mint &a) { return os << a.x; }

using edge = struct {
  ll to;
  ll cost;
};

struct fpoint {
  ld x;
  ld y;
  bool operator<(const fpoint &p) const {
    if (x == p.x)
      return y < p.y;
    return x < p.x;
  }

  bool operator==(const fpoint &p) const {
    if (p.x - EPS < x && x < p.x + EPS && p.y - EPS < y && y < p.y + EPS)
      return true;
    return false;
  }

  bool operator!=(const fpoint &p) const {
    if (p.x - EPS > x || x > p.x + EPS || p.y - EPS > y || y > p.y + EPS)
      return true;
    return false;
  }

  fpoint &operator+=(const ld a) {
    x += a, y += a;
    return *this;
  }

  fpoint &operator-=(const ld a) {
    x -= a, y -= a;
    return *this;
  }

  fpoint &operator*=(const ld a) {
    x *= a, y *= a;
    return *this;
  }

  fpoint &operator/=(const ld a) {
    x /= a, y /= a;
    return *this;
  }

  fpoint &operator+=(const fpoint p) {
    x += p.x, y += p.y;
    return *this;
  }

  fpoint &operator-=(const fpoint p) {
    x -= p.x, y -= p.y;
    return *this;
  }

  fpoint &operator*=(const fpoint p) {
    x *= p.x, y *= p.y;
    return *this;
  }

  fpoint &operator/=(const fpoint p) {
    x /= p.x, y /= p.y;
    return *this;
  }

  bool is_zero() {
    fpoint z = {0, 0};
    return z == *this;
  }

  fpoint operator+(const fpoint p) const { return fpoint(*this) += p; }
  fpoint operator-(const fpoint p) const { return fpoint(*this) -= p; }
  fpoint operator*(const fpoint p) const { return fpoint(*this) *= p; }
  fpoint operator/(const fpoint p) const { return fpoint(*this) /= p; }
  fpoint operator+(const ld a) const { return fpoint(*this) += a; }
  fpoint operator-(const ld a) const { return fpoint(*this) -= a; }
  fpoint operator*(const ld a) const { return fpoint(*this) *= a; }
  fpoint operator/(const ld a) const { return fpoint(*this) /= a; }
};

struct point {
  ll x;
  ll y;
  bool operator<(const point &p) const {
    if (x == p.x)
      return y < p.y;
    return x < p.x;
  }

  bool operator==(const point &p) const {
    if (x == p.x && y == p.y)
      return true;
    return false;
  }

  bool operator!=(const point &p) const {
    if (x != p.x || y != p.y)
      return true;
    return false;
  }

  point &operator+=(const ll a) {
    x += a, y += a;
    return *this;
  }

  point &operator-=(const ll a) {
    x -= a, y -= a;
    return *this;
  }

  point &operator*=(const ll a) {
    x *= a, y *= a;
    return *this;
  }

  point &operator+=(const point p) {
    x += p.x, y += p.y;
    return *this;
  }

  point &operator-=(const point p) {
    x -= p.x, y -= p.y;
    return *this;
  }

  point &operator*=(const point p) {
    x *= p.x, y *= p.y;
    return *this;
  }

  bool is_zero() {
    point z = {0, 0};
    return z == *this;
  }

  fpoint to_fpoint() {
    fpoint ret = {ld(x), ld(y)};
    return ret;
  }

  void operator++(int) { x++, y++; }
  void operator++() { x++, y++; }
  void operator--(int) { x--, y--; }
  void operator--() { x--, y--; }
  point operator+(const point p) const { return point(*this) += p; }
  point operator-(const point p) const { return point(*this) -= p; }
  point operator*(const point p) const { return point(*this) *= p; }
  point operator+(const ll a) const { return point(*this) += a; }
  point operator-(const ll a) const { return point(*this) -= a; }
  point operator*(const ll a) const { return point(*this) *= a; }
};

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::tuple<ll, ll, ll> tp3;
typedef std::tuple<ll, ll, ll, ll> tp4;
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<vl> vl2;
typedef std::vector<vl2> vl3;
typedef std::vector<vl3> vl4;
typedef std::vector<mint> vmi;
typedef std::vector<vmi> vmi2;
typedef std::vector<vmi2> vmi3;
typedef std::vector<vmi3> vmi4;
typedef std::vector<bool> vb;
typedef std::vector<vb> vb2;
typedef std::vector<vb2> vb3;
typedef std::vector<vb3> vb4;
typedef std::vector<pl> vpl;
typedef std::vector<tp3> vtp3;
typedef std::vector<tp4> vtp4;
typedef std::vector<point> points;
typedef std::vector<fpoint> fpoints;
// priority queue. Taking from the higher value. Don't forget calling !q.empty()
typedef std::priority_queue<ll> pq;
// priority queue. Taking from the lower value
typedef std::priority_queue<ll, vl, greater<ll>> pql;
typedef std::vector<vector<edge>> Graph;
const ll N_DIGITS = 60;
const long double PI = 3.14159265358979323846;
points 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(mint a) { return to_string(a.x); }

string to_string(point p) {
  return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";
}

string to_string(fpoint p) {
  return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";
}

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 mt make_tuple
#define pr(a) std::cout << (a)
#define prl(a) std::cout << (a) << endl
#define prl2(a, b) std::cout << (a) << " " << (b) << endl
#define prl3(a, b, c) std::cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a, b, c, d)                                                       \\
  std::cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) std::cout << (a) << " "
#define prs2(a, b) std::cout << (a) << " " << (b) << " "
#define prs3(a, b, c) std::cout << (a) << " " << (b) << " " << (c) << " "
#define prs4(a, b, c, d)                                                       \\
  std::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 in5(a, b, c, d, e) cin >> (a) >> (b) >> (c) >> (d) >> (e)
#define in6(a, b, c, d, e, f) cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f)
#define in7(a, b, c, d, e, f, g)                                               \\
  cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f) >> (g)
#define e1 first
#define e2 second
#define ltos(n) to_string((n))
#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)                                                    \\
  std::cout << fixed << setprecision(digits) << (num) << endl;
#define prl_time(s)                                                            \\
  prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))

#ifdef _LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif

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) {
    // complexity O(EV^2)
    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;
    }
  }
};

struct UnionFind {
  vl parents, set_size;
  set<ll> root_idx;
  ll n_groups;

  UnionFind(ll n) {
    parents = set_size = vl(n);
    n_groups = n;
    rep(i, n) {
      parents[i] = i;
      set_size[i] = 1LL;
      root_idx.insert(i);
    }
  }

  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];
    root_idx.erase(y);
    n_groups--;
  }

  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)]; }

  ll num_union() const { return n_groups; }
};

struct Doubling { // ABC167D
  ll n;
  ll sz;
  vl2 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 LowestCommonAncestor {
public:
  ll N, logN;
  vl depth, len;
  Graph tree;
  vl2 parents;

  LowestCommonAncestor(ll n, ll offset = -1, bool is_weighted = true) {
    N = n;
    logN = 0;
    while (N > (1LL << logN))
      logN++;
    depth = len = vl(N);
    parents = vl2(logN, vl(N));
    tree = Graph(N);
    input(N, offset, is_weighted);
    init(0, -1, 0, 0);
    build();
  }

  void add_edge(ll from, ll to, ll dist) {
    tree[from].pb({to, dist});
    tree[to].pb({from, dist});
  }

  void input(ll n, ll offset = -1, bool is_weighted = true) {
    rep(i, n - 1) {
      ll a, b, d = 1;
      in2(a, b);
      a += offset, b += offset;
      if (is_weighted)
        in1(d);
      add_edge(a, b, d);
    }
  }

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

  //       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, d);
    // 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)
    */

    if (i < 0)
      return ini;
    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) {
    // a[l] + a[l + 1] + ... + a[r - 1] + a[r]
    return query(r) - query(l - 1);
  }

  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() {
    vb 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(); }
};

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

  // This is the exaple of the number of children
  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;
  }
};

struct StronglyConnectedComponents {
  ll n, n_cmp;
  vl2 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 n_cmp = num;
  }
};

struct CombinationMemo {
  ll sz, mod;
  vl facts, facts_inv, minv;

  CombinationMemo(ll sz, ll _mod) : sz(sz), mod(_mod) {
    facts.resize(sz + 5);
    facts_inv.resize(sz + 5);
    minv.resize(sz + 5);
    init();
  }

  void init() {
    facts[0] = facts[1] = 1;
    minv[1] = 1;
    facts_inv[0] = facts_inv[1] = 1;

    For(i, 2, sz + 3) {
      facts[i] = (i * facts[i - 1]) % mod;
      minv[i] = mod - minv[mod % i] * (mod / i) % mod;
      facts_inv[i] = facts_inv[i - 1] * minv[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;
  }

  ll nPk(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;
    return val % mod;
  }
};

struct PowerMemo {
  ll sz, mod, base;
  vl powB;

  PowerMemo(ll sz, ll base, ll _mod) : sz(sz), base(base), mod(_mod) {
    powB.resize(sz + 5);
    init();
  }

  void init() {
    powB[0] = 1;
    rep(i, sz + 3) powB[i + 1] = (powB[i] * base) % mod;
  }

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

struct Grid2D {
  Graph graph;
  ll Width, Height;

  Grid2D(ll w, ll h) : Width(w), Height(h) { graph.resize(w * h); }

  ll pos_to_idx(point p) { return p.y * Width + p.x; }

  point idx_to_pos(ll idx) { return {idx % Width, idx / Width}; }

  bool undefined_region(point p, vb2 &block) {
    if (p.x < 0 || p.x > Width - 1)
      return true;
    if (p.y < 0 || p.y > Height - 1)
      return true;
    if (block[p.x][p.y])
      return true;
    return false;
  }

  void build(vb2 &block, ll val = 1) {
    rep(x, Width) rep(y, Height) {
      point p = {x, y};
      ll idx1 = pos_to_idx(p);
      ll idx2;

      if (block[x][y])
        continue;
      rep(i, 4) {
        point nxt = p + dirs[i];
        idx2 = pos_to_idx(nxt);
        if (!undefined_region(nxt, block))
          graph[idx1].pb(
              {idx2, val}); // dist[idx1][idx2] = val; (warshall-floyd)
      }
    }
  }
};

struct Cumulative2D {
  vl2 cum;
  ll w, h;

  Cumulative2D(ll w, ll h) : w(w), h(h) { cum = vl2(w + 1, vl(h + 1, 0)); }

  template <typename T> void build(vector<T> &vec) {
    // never forget building
    rep(x, w + 1) cum[x][0] = 0;
    rep(y, h + 1) cum[0][y] = 0;
    rep(y, h) rep(x, w) cum[x + 1][y + 1] = cum[x][y + 1] + vec[x][y];
    rep(x, w + 1) rep(y, h) cum[x][y + 1] += cum[x][y];
  }

  ll func(ll x, ll y, ll dx, ll dy) {
    // 1-indexed
    // the rectangle of (x, y), (x + dx, y), (x, y + dy) and (x + dx, y + dy)
    // think about the case of (1, 1, 1, 1).
    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;
  }
};

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_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 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 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;
}

vl2 pascal_triangle(ll n) {
  /*
  Complexity: O(n^2)
  The upper bound of n is nearly 50.

  Parameters
  ----------
  n; the size of returned vector

  Returns
  -------
  comb[i][j]: combination(i, j). 0 <= i <= n, 0 <= j <= i
  */

  vl2 comb(n + 1, vl(n + 1));
  comb[0][0] = 1;
  For(i, 1, n + 1) rep(j, i + 1) {
    comb[i][j] += comb[i - 1][j];
    if (j > 0)
      comb[i][j] += comb[i - 1][j - 1];
  }

  return comb;
}

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;
}

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

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;
}

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 next_combination(ll sub) {
  /*
  nCk
  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;

  if (x != 0)
    return (((sub & ~y) / x) >> 1) | y;
  else
    return INF;
}

// 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
template <typename T> vl z_algorithm(T &s, ll n) {
  /*
  Paramters
  ---------
  T: the string or list of interest
  n: the size of string or list

  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.
  */
  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 = s[l - 1 - idx] - '0';
    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 = s[l - 1 - idx] - '0';
    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;
}

template <typename T> bool isColinear(T p1, T p2, T p3) {
  T v1 = p2 - p1, v2 = p3 - p1;
  return v1.x * v2.y == v1.y * v2.x;
}

template <typename T> T PerpendicularBisector(T p1, T p2) {
  T vec = p2 - p1;
  assert(!vec.is_zero());
  T ret = {vec.y, -vec.x};

  return ret;
}

template <typename T> ld Distance2DPoints(T p1, T p2) {
  T vec = (p1 - p2) * (p1 - p2);
  return sqrt(vec.x + vec.y);
}

ll SquaredDistance2DPoints(point p1, point p2) {
  point vec = (p1 - p2) * (p1 - p2);
  return vec.x + vec.y;
}

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> OuterCenter(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
  point ret;

  if (isColinear(p1, p2, p3)) {
    ll d1 = SquaredDistance2DPoints(p1, p2);
    ll d2 = SquaredDistance2DPoints(p2, p3);
    ll d3 = SquaredDistance2DPoints(p3, p1);
    if (d1 >= d2 && d1 >= d3) {
      ret = p1 + p2;
      return mp(ret, 2);
    } else if (d2 >= d1 && d2 >= d3) {
      ret = p2 + p3;
      return mp(ret, 2);
    } else {
      ret = p3 + p1;
      return mp(ret, 2);
    }
  }

  point pv1 = PerpendicularBisector(p1, p2);
  point pv2 = PerpendicularBisector(p1, p3);
  point cv1_2x = p1 + p2, cv2_2x = p1 + p3;
  // cv1 + k pv1 == cv2 + m pv2
  // (pv1x -pv2x) (k) = (cv2x - cv1x)
  // (pv1y -pv2y) (m) = (cv2y - cv1y)
  ll det_inv = -pv1.x * pv2.y + pv1.y * pv2.x;
  ll x1_2x = cv2_2x.x - cv1_2x.x, x2_2x = cv2_2x.y - cv1_2x.y;
  pl c_2x_det = {-pv2.y * x1_2x + pv2.x * x2_2x,
                 -pv1.y * x1_2x + pv1.x * x2_2x};

  // ret.x = ld(cv1_2x.x * det_inv + pv1.x * c_2x_det.e1) / ld(2 * det_inv);
  // ret.y = ld(cv1_2x.y * det_inv + pv1.y * c_2x_det.e1) / ld(2 * det_inv);
  ret.x = cv1_2x.x * det_inv + pv1.x * c_2x_det.e1;
  ret.y = cv1_2x.y * det_inv + pv1.y * c_2x_det.e1;

  ll jacobian = 2 * det_inv;
  return mp(ret, jacobian);
}

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.
  */

  BinaryIndexedTree 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;
}

ld bin_search(ld left, ld right, bool lb, function<bool(ld)> judge) {
  ld mid;
  while (right - left > EPS) {
    mid = (right + left) / 2;
    if (lb) {
      if (judge(mid))
        right = mid;
      else
        left = mid + EPS;
    } else {
      if (judge(mid))
        left = mid;
      else
        right = mid - EPS;
    }
  }
  return right;
}

ll bin_search(ll left, ll right, bool lb, function<bool(ll)> judge) {
  ll mid;
  while (right > left) {
    if (lb) {
      // if true (satisfies the condition), range shifts smaller direction
      mid = (right + left) / 2;
      if (judge(mid))
        right = mid;
      else
        left = mid + 1;
    } else {
      // if true (satisfies the condition), range shitfs larger direction
      mid = (right + left + 1) / 2;
      if (judge(mid))
        left = mid;
      else
        right = mid - 1;
    }
  }
  return right;
}

ld trinary_search(ld left, ld right, function<ld(ld)> func) {
  // Care the value EPS!!! Compare to the condition
  while (abs(right - left) > EPS) {
    ld left2 = (2.0 * left + right) / 3.0;
    ld right2 = (left + 2.0 * right) / 3.0;
    ld f1 = func(left2);
    ld f2 = func(right2);
    if (f1 <= f2)
      right = right2;
    else if (f2 <= f1)
      left = left2;
  }

  return right;
}

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) {
  // O(ElogE)
  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 = v.size();
  vl dp(n, INF);
  rep(i, n) * lower_bound(all(dp), v[i]) = v[i];
  return lower_bound(all(dp), INF) - dp.begin();
}

void dijkstra(ll start, Graph &graph, vl &dist, vl &vertex_pre,
              bool trace = false) {
  priority_queue<pl, vpl, greater<pl>> edge_costs;
  ll n = graph.size();
  dist = vl(n, INF);
  if (trace)
    vertex_pre = vl(n, -1);
  dist[start] = 0;
  edge_costs.push(pl(0, start));

  while (!edge_costs.empty()) {
    ll idx, cost;
    tie(cost, idx) = edge_costs.top();
    edge_costs.pop();
    if (dist[idx] < cost)
      continue;

    for (auto e : graph[idx]) {
      if (dist[e.to] > dist[idx] + e.cost) {
        dist[e.to] = dist[idx] + e.cost;
        if (trace)
          vertex_pre[e.to] = idx;
        edge_costs.push(pl(dist[e.to], e.to));
      }
    }
  }
}

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

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

void warshall_floyd(vl2 &dist) {
  ll n = dist.size();
  // Dont forget the initialization
  // 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]);
}

// ABC061D
bool find_negative_cycle(ll n, ll goal, Graph &graph, vl &dist) {
  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 goal, Graph &graph, vl &dist) {
  // 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)
  ll n = graph.size();
  dist = vl(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(n, goal, graph, dist))
    return false;
  return true;
}

void Euler_Tour(Graph &tree, vl &euler_tour, vl &in, vl &out, ll anc = 0) {
  // record only when we first reach a node and leave the node.
  ll n = tree.size();
  vb visited(n, false);
  in = out = vl(n);

  auto dfs = [&](auto dfs, ll source) -> void {
    visited[source] = true;
    in[source] = euler_tour.size();
    euler_tour.pb(source);

    bool is_leaf = true;
    for (auto &e : tree[source]) {
      ll target = e.to;
      if (visited[target])
        continue;
      else
        is_leaf = false;
      dfs(dfs, target);
    }

    euler_tour.pb(-source);
    out[source] = euler_tour.size() - 1;
  };

  dfs(dfs, anc);
}

void Euler_Tour2(Graph &tree, vl &euler_tour, vl &in, vl &out, ll anc = 0) {
  // record everytime we reach a node
  ll n = tree.size();
  vb visited(n, false);
  in = out = vl(n);

  auto dfs = [&](auto dfs, ll source) -> void {
    visited[source] = true;
    in[source] = euler_tour.size();
    euler_tour.pb(source);

    bool is_leaf = true;
    for (auto &e : tree[source]) {
      ll target = e.to;
      if (visited[target])
        continue;
      else
        is_leaf = false;
      dfs(dfs, target);
      euler_tour.pb(source);
    }

    out[source] = euler_tour.size() - 1;
  };

  dfs(dfs, anc);
}

std::mt19937 create_rand_engine() {
  std::random_device rnd;
  std::vector<std::uint_least32_t> v(10);
  std::generate(v.begin(), v.end(), std::ref(rnd));
  std::seed_seq seed(v.begin(), v.end());
  return std::mt19937(seed);
}

vl generate_unique_array(const size_t sz, ll vm, ll vM) {
  const size_t range = static_cast<size_t>(vM - vm + 1);
  const size_t num = static_cast<size_t>(sz * 1.2);
  assert(vm <= vM);
  assert(sz <= range);

  vl ret;
  auto engine = create_rand_engine();
  std::uniform_int_distribution<ll> distribution(vm, vM);

  while (ret.size() < sz) {
    while (ret.size() < num)
      ret.pb(distribution(engine));
    sort(all(ret));

    auto unique_end = unique(all(ret));
    if (sz < distance(ret.begin(), unique_end))
      unique_end = next(ret.begin(), sz);

    ret.erase(unique_end, ret.end());
  }

  return ret;
}

vl generate_array(const size_t sz, ll vm, ll vM) {
  const size_t range = static_cast<size_t>(vM - vm + 1);
  assert(vm <= vM);

  vl ret;
  auto engine = create_rand_engine();
  std::uniform_int_distribution<ll> distribution(vm, vM);

  while (ret.size() < sz)
    ret.pb(distribution(engine));

  return ret;
}

void read_graph(ll n_vertex, ll n_edges, Graph &graph, ll cost = -1,
                bool directed = false, ll offset = -1) {

  graph.resize(n_vertex);
  rep(i, n_edges) {
    ll from, to, c = cost;
    if (cost == -1)
      in3(from, to, c);
    else
      in2(from, to);
    from += offset, to += offset;

    graph[from].pb({to, cost});
    if (!directed)
      graph[to].pb({from, cost});
  }
}

void read_vector(ll n, vl &v, ll offset = 0) {
  v.resize(n);
  rep(i, n) {
    in1(v[i]);
    v[i] += offset;
  }
}

void read_pair_vector(ll n, vpl &v, ll offset1 = 0, ll offset2 = 0,
                      bool rev = false) {
  v.resize(n);
  rep(i, n) {
    ll a, b;
    in2(a, b);
    a += offset1, b += offset2;
    if (!rev)
      v[i] = {a, b};
    else
      v[i] = {b, a};
  }
}

void read_2DMap(ll w, ll h, vb2 &block, char b) {
  block = vb2(w, vb(h, false));
  str s;
  rep(y, h) {
    in1(s);
    rep(x, w) block[x][y] = (s[x] == b);
  }
}

/*

diameter of tree
Graph tree;
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

*/

/*

    # 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));

    # 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){
        if (l == r) r++;
        while(r < n && cond) r++;
        l++;
    }
    prl(answer);

    #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);
    }
*/

/*
     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

####### Attention #######
S & (1 << i) -> if true, i in S
S | (1 << i) -> S union {i}
S & ~(1 << i) -> S - {i}
__builtin_popcountl(i) -> the number of 1 in binary

#Conditional Operator
condition ? true : false;

# 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;
const size_t MAX_BIT = 100;
typedef bitset<MAX_BIT> bts;
typedef vector<bts> vbt;
typedef vector<vbt> vbt2;
typedef vector<vbt2> vbt3;
bool okay = false;
ll answer = 0;
str s, t, u;
ll n, m, h, w, k, Q;
ll a, b, c, d, idx, cnt;
vl as, bs, cs, ds;
vb2 block;
vb visited;
Graph graph;
Graph tree;
/*
read_graph(n, m, graph, cost, directed, offset)
    cost: if link cost is fixed, put positive integer
          otherwise -1
    directed: if directed true, olse false
    offset: if the given vertex is 1-indexed, -1.
*/

void solve() {
  // generate_array(size, min, max);
  in1(n);
  priority_queue<tp3> q, nxt;
  ll tot = 0, lb = 0, rb = 0;
  rep(i, n) {
    in1(s);
    ll r1 = 0, r2 = 0, l1 = 0, l2 = 0;
    m = s.length();
    ll v1 = 0, v2 = 0, v3 = 0;
    rep(j, m) {
      if (s[j] == '(')
        l1++;
      else
        r1++;
      if (s[m - 1 - j] == '(')
        l2++;
      else
        r2++;
      chmin(v1, l1 - r1);
      chmin(v3, r2 - l2);
    }
    v2 = l1 - r1;
    tot += v2;
    if (v1 >= 0)
      lb += v2;
    else if (v3 >= 0)
      rb -= v2;
    else
      q.push(mt(v1, v2, v3));
  }

  if (tot != 0) {
    prl("No");
    return;
  }
  while (!q.empty()) {
    ll v1, v2, v3;
    tie(v1, v2, v3) = q.top();
    q.pop();
    if (lb + v1 >= 0 && v2 >= 0)
      lb += v2;
    else
      nxt.push(mt(v3, -v2, v1));
  }
  swap(q, nxt);

  while (!q.empty()) {
    ll v1, v2, v3;
    tie(v1, v2, v3) = q.top();
    q.pop();
    if (rb + v1 >= 0 && v2 >= 0)
      rb += v2;
    else
      nxt.push(mt(v3, -v2, v1));
  }
  swap(q, nxt);

  ll sz = q.size();
  assert(sz <= 0);
  while (!q.empty()) {
    ll v1, v2, v3;
    tie(v1, v2, v3) = q.top();
    q.pop();
    if (lb + v1 >= 0)
      lb += v2;
    else {
      prl("No");
      return;
    }
  }
  prl("Yes");

  auto naive_solver = [&]() {};
#ifdef _LOCAL
  naive_solver();
#endif
}

void test(ll num = 0, bool verbose = false) {
  rep(i, max(1LL, num)) {
    ll t = clock();
    if (verbose)
      prl4("\\n#####", i + 1, "#####", "\\n## Answer ##");
    solve();
    if (verbose) {
      prl("");
      prl_time(t);
    }
  }
}

int main(void) {
#ifdef _LOCAL
  test(4, true);
#else
  test(0, false);
#endif

  return 0;
}