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

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

#include <bits/stdc++.h>

using namespace std;

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

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

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

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

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

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

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

#define stoi stoll

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  bool is_prepared() { return prepared; }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 4 "/home/maspy/compro/library/graph/tree.hpp"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  // path [a,b] と [c,d] の交わり. 空ならば {-1,-1}.
  // https://codeforces.com/problemset/problem/500/G
  pair<int, int> path_intersection(int a, int b, int c, int d) {
    int ab = lca(a, b), ac = lca(a, c), ad = lca(a, d);
    int bc = lca(b, c), bd = lca(b, d), cd = lca(c, d);
    int x = ab ^ ac ^ bc, y = ab ^ ad ^ bd; // meet(a,b,c), meet(a,b,d)
    if (x != y) return {x, y};
    int z = ac ^ ad ^ cd;
    if (x != z) x = -1;
    return {x, x};
  }
};
#line 2 "/home/maspy/compro/library/ds/splaytree/splaytree.hpp"

/*
update でちゃんと prod が計算されてくれれば prod は op(lprod,x,rprod) でなくてもよい.
*/

// Node 型を別に定義して使う
template <typename Node>
struct SplayTree {
  Node *pool;
  const int NODES;
  int pid;
  using np = Node *;
  using X = typename Node::value_type;
  using A = typename Node::operator_type;
  vc<np> FREE;

  SplayTree(int NODES) : NODES(NODES), pid(0) { pool = new Node[NODES]; }
  ~SplayTree() { delete[] pool; }

  void free_subtree(np c) {
    if (!c) return;
    auto dfs = [&](auto &dfs, np c) -> void {
      if (c->l) dfs(dfs, c->l);
      if (c->r) dfs(dfs, c->r);
      FREE.eb(c);
    };
    dfs(dfs, c);
  }

  void reset() {
    pid = 0;
    FREE.clear();
  }

  np new_root() { return nullptr; }

  np new_node(const X &x) {
    assert(!FREE.empty() || pid < NODES);
    np n = (FREE.empty() ? &(pool[pid++]) : POP(FREE));
    Node::new_node(n, x);
    return n;
  }

  np new_node(const vc<X> &dat) {
    auto dfs = [&](auto &dfs, int l, int r) -> np {
      if (l == r) return nullptr;
      if (r == l + 1) return new_node(dat[l]);
      int m = (l + r) / 2;
      np l_root = dfs(dfs, l, m);
      np r_root = dfs(dfs, m + 1, r);
      np root = new_node(dat[m]);
      root->l = l_root, root->r = r_root;
      if (l_root) l_root->p = root;
      if (r_root) r_root->p = root;
      root->update();
      return root;
    };
    return dfs(dfs, 0, len(dat));
  }

  u32 get_size(np root) { return (root ? root->size : 0); }

  np merge(np l_root, np r_root) {
    if (!l_root) return r_root;
    if (!r_root) return l_root;
    assert((!l_root->p) && (!r_root->p));
    splay_kth(r_root, 0); // splay したので prop 済
    r_root->l = l_root;
    l_root->p = r_root;
    r_root->update();
    return r_root;
  }
  np merge3(np a, np b, np c) { return merge(merge(a, b), c); }
  np merge4(np a, np b, np c, np d) { return merge(merge(merge(a, b), c), d); }

  pair<np, np> split(np root, u32 k) {
    assert(!root || !root->p);
    if (k == 0) return {nullptr, root};
    if (k == (root->size)) return {root, nullptr};
    splay_kth(root, k - 1);
    np right = root->r;
    root->r = nullptr, right->p = nullptr;
    root->update();
    return {root, right};
  }
  tuple<np, np, np> split3(np root, u32 l, u32 r) {
    np nm, nr;
    tie(root, nr) = split(root, r);
    tie(root, nm) = split(root, l);
    return {root, nm, nr};
  }
  tuple<np, np, np, np> split4(np root, u32 i, u32 j, u32 k) {
    np d;
    tie(root, d) = split(root, k);
    auto [a, b, c] = split3(root, i, j);
    return {a, b, c, d};
  }

  // 部分木が区間 [l,r) に対応するようなノードを作って返す
  // そのノードが root になるわけではないので、
  // このノードを参照した後にすぐに splay して根に持ち上げること
  void goto_between(np &root, u32 l, u32 r) {
    if (l == 0 && r == root->size) return;
    if (l == 0) {
      splay_kth(root, r);
      root = root->l;
      return;
    }
    if (r == root->size) {
      splay_kth(root, l - 1);
      root = root->r;
      return;
    }
    splay_kth(root, r);
    np rp = root;
    root = rp->l;
    root->p = nullptr;
    splay_kth(root, l - 1);
    root->p = rp;
    rp->l = root;
    rp->update();
    root = root->r;
  }

  vc<X> get_all(const np &root) {
    vc<X> res;
    auto dfs = [&](auto &dfs, np root) -> void {
      if (!root) return;
      root->prop();
      dfs(dfs, root->l);
      res.eb(root->get());
      dfs(dfs, root->r);
    };
    dfs(dfs, root);
    return res;
  }

  X get(np &root, u32 k) {
    assert(root == nullptr || !root->p);
    splay_kth(root, k);
    return root->get();
  }

  void set(np &root, u32 k, const X &x) {
    assert(root != nullptr && !root->p);
    splay_kth(root, k);
    root->set(x);
  }

  void multiply(np &root, u32 k, const X &x) {
    assert(root != nullptr && !root->p);
    splay_kth(root, k);
    root->multiply(x);
  }

  X prod(np &root, u32 l, u32 r) {
    assert(root == nullptr || !root->p);
    using Mono = typename Node::Monoid_X;
    if (l == r) return Mono::unit();
    assert(0 <= l && l < r && r <= root->size);
    goto_between(root, l, r);
    X res = root->prod;
    splay(root, true);
    return res;
  }

  X prod(np &root) {
    assert(root == nullptr || !root->p);
    using Mono = typename Node::Monoid_X;
    return (root ? root->prod : Mono::unit());
  }

  void apply(np &root, u32 l, u32 r, const A &a) {
    if (l == r) return;
    assert(0 <= l && l < r && r <= root->size);
    goto_between(root, l, r);
    root->apply(a);
    splay(root, true);
  }
  void apply(np &root, const A &a) {
    if (!root) return;
    root->apply(a);
  }

  void reverse(np &root, u32 l, u32 r) {
    assert(root == nullptr || !root->p);
    if (l == r) return;
    assert(0 <= l && l < r && r <= root->size);
    goto_between(root, l, r);
    root->reverse();
    splay(root, true);
  }
  void reverse(np root) {
    if (!root) return;
    root->reverse();
  }

  void rotate(Node *n) {
    // n を根に近づける。prop, update は rotate の外で行う。
    Node *pp, *p, *c;
    p = n->p;
    pp = p->p;
    if (p->l == n) {
      c = n->r;
      n->r = p;
      p->l = c;
    } else {
      c = n->l;
      n->l = p;
      p->r = c;
    }
    if (pp && pp->l == p) pp->l = n;
    if (pp && pp->r == p) pp->r = n;
    n->p = pp;
    p->p = n;
    if (c) c->p = p;
  }

  void prop_from_root(np c) {
    if (!c->p) {
      c->prop();
      return;
    }
    prop_from_root(c->p);
    c->prop();
  }

  void splay(Node *me, bool prop_from_root_done) {
    // これを呼ぶ時点で、me の祖先（me を除く）は既に prop 済であることを仮定
    // 特に、splay 終了時点で me は upd / prop 済である
    if (!prop_from_root_done) prop_from_root(me);
    me->prop();
    while (me->p) {
      np p = me->p;
      np pp = p->p;
      if (!pp) {
        rotate(me);
        p->update();
        break;
      }
      bool same = (p->l == me && pp->l == p) || (p->r == me && pp->r == p);
      if (same) rotate(p), rotate(me);
      if (!same) rotate(me), rotate(me);
      pp->update(), p->update();
    }
    // me の update は最後だけでよい
    me->update();
  }

  void splay_kth(np &root, u32 k) {
    assert(0 <= k && k < (root->size));
    while (1) {
      root->prop();
      u32 sl = (root->l ? root->l->size : 0);
      if (k == sl) break;
      if (k < sl)
        root = root->l;
      else {
        k -= sl + 1;
        root = root->r;
      }
    }
    splay(root, true);
  }

  // check(x), 左側のノード全体が check を満たすように切る
  template <typename F>
  pair<np, np> split_max_right(np root, F check) {
    if (!root) return {nullptr, nullptr};
    assert(!root->p);
    np c = find_max_right(root, check);
    if (!c) {
      splay(root, true);
      return {nullptr, root};
    }
    splay(c, true);
    np right = c->r;
    if (!right) return {c, nullptr};
    right->p = nullptr;
    c->r = nullptr;
    c->update();
    return {c, right};
  }

  // check(x, cnt), 左側のノード全体が check を満たすように切る
  template <typename F>
  pair<np, np> split_max_right_cnt(np root, F check) {
    if (!root) return {nullptr, nullptr};
    assert(!root->p);
    np c = find_max_right_cnt(root, check);
    if (!c) {
      splay(root, true);
      return {nullptr, root};
    }
    splay(c, true);
    np right = c->r;
    if (!right) return {c, nullptr};
    right->p = nullptr;
    c->r = nullptr;
    c->update();
    return {c, right};
  }

  // 左側のノード全体の prod が check を満たすように切る
  template <typename F>
  pair<np, np> split_max_right_prod(np root, F check) {
    if (!root) return {nullptr, nullptr};
    assert(!root->p);
    np c = find_max_right_prod(root, check);
    if (!c) {
      splay(root, true);
      return {nullptr, root};
    }
    splay(c, true);
    np right = c->r;
    if (!right) return {c, nullptr};
    right->p = nullptr;
    c->r = nullptr;
    c->update();
    return {c, right};
  }

  template <typename F>
  np find_max_right(np root, const F &check) {
    // 最後に見つけた ok の点、最後に探索した点
    np last_ok = nullptr, last = nullptr;
    while (root) {
      last = root;
      root->prop();
      if (check(root->x)) {
        last_ok = root;
        root = root->r;
      } else {
        root = root->l;
      }
    }
    splay(last, true);
    return last_ok;
  }

  template <typename F>
  np find_max_right_cnt(np root, const F &check) {
    // 最後に見つけた ok の点、最後に探索した点
    np last_ok = nullptr, last = nullptr;
    ll n = 0;
    while (root) {
      last = root;
      root->prop();
      ll ns = (root->l ? root->l->size : 0);
      if (check(root->x, n + ns + 1)) {
        last_ok = root;
        n += ns + 1;
        root = root->r;
      } else {
        root = root->l;
      }
    }
    splay(last, true);
    return last_ok;
  }

  template <typename F>
  np find_max_right_prod(np root, const F &check) {
    using Mono = typename Node::Monoid_X;
    X prod = Mono::unit();
    // 最後に見つけた ok の点、最後に探索した点
    np last_ok = nullptr, last = nullptr;
    while (root) {
      last = root;
      root->prop();
      np tmp = root->r;
      root->r = nullptr;
      root->update();
      X lprod = Mono::op(prod, root->prod);
      root->r = tmp;
      root->update();
      if (check(lprod)) {
        prod = lprod;
        last_ok = root;
        root = root->r;
      } else {
        root = root->l;
      }
    }
    splay(last, true);
    return last_ok;
  }
};
#line 2 "/home/maspy/compro/library/alg/monoid/add_pair.hpp"

template <typename E>
struct Monoid_Add_Pair {
  using value_type = pair<E, E>;
  using X = value_type;
  static constexpr X op(const X &x, const X &y) {
    return {x.fi + y.fi, x.se + y.se};
  }
  static constexpr X inverse(const X &x) { return {-x.fi, -x.se}; }
  static constexpr X unit() { return {0, 0}; }
  static constexpr bool commute = true;
};
#line 3 "/home/maspy/compro/library/convex/slope_super.hpp"

namespace SLOPE_TRICK_SUPER {
/*
傾きと座標が全部 T.
(x0,y0,a0) / 傾き変化を splay tree で持つ.
末尾には必ず infty が入っているようにする.
(0,10),(1,6),(3,4),(6,7)
->
(x0,y0,a0)=(0,10,-4)
dat = ([1,3],[3,2])

f(x) の計算, (min,argmin) の計算
加法, 畳み込み

加法: easy
f(x) の計算: sum(a), sum(xa) が要る
畳み込み: x->x+c が要る
*/

template <typename T>
struct Node {
  using value_type = pair<T, T>;
  using operator_type = T;
  using np = Node *;
  using Monoid_X = Monoid_Add_Pair<T>;

  np p, l, r;
  bool rev;
  u32 size;
  pair<T, T> x;    // (x,a)
  pair<T, T> prod; // (a sum, xa sum)
  T add_x;

  static void new_node(np n, const pair<T, T> &x) {
    n->p = n->l = n->r = nullptr, n->rev = 0, n->size = 1;
    n->x = x, n->prod = {x.se, x.fi * x.se}, n->add_x = 0;
  }

  void update() {
    size = 1;
    if (l) { size += l->size; }
    if (r) { size += r->size; }
    prod = {x.se, x.fi * x.se};
    if (l) prod = Monoid_X::op(prod, l->prod);
    if (r) prod = Monoid_X::op(prod, r->prod);
  }

  void prop() {
    assert(!rev);
    if (add_x == 0) return;
    if (l) l->x.fi += add_x, l->prod.se += l->prod.fi * add_x, l->add_x += add_x;
    if (r) r->x.fi += add_x, r->prod.se += r->prod.fi * add_x, r->add_x += add_x;
    add_x = 0;
  }

  void apply(T a) { x.fi += a, prod.se += a * prod.fi, add_x += a; }

  // update, prop 以外で呼ばれるものは、splay 後であることが想定されている。
  // したがってその時点で update, prop 済であることを仮定してよい。
  pair<T, T> get() { return x; }
  void set(const pair<T, T> &xx) {
    x = xx;
    update();
  }
};

// 関数は破壊的な変更にされる
template <typename T>
struct Slope_Trick_Super {
  SplayTree<Node<T>> ST;
  using np = Node<T> *;

  struct FUNC {
    np root; // 定義域がこわれていたら root == empty
    T x0, x1, a0, y0;
    int size() { return (root ? root->size : 0); }
  };

  Slope_Trick_Super(int NODES) : ST(NODES) {}

  // (L,R,a,b) : [L,R] で y=ax+b
  FUNC segment_func(T L, T R, T a, T b) { return {nullptr, L, R, a, a * L + b}; }
  FUNC from_points(vc<pair<T, T>> &point) {
    return from_points(len(point), [&](int i) -> pair<T, T> { return point[i]; });
  }
  template <typename F>
  FUNC from_points(int N, F f) {
    vc<T> X(N), Y(N);
    FOR(i, N) tie(X[i], Y[i]) = f(i);
    if (N == 1) return segment_func(X[0], X[0], 0, Y[0]);
    T a0 = (Y[1] - Y[0]) / (X[1] - X[0]);
    T x0 = X[0], x1 = X.back();
    vc<pair<T, T>> dat;
    T a = a0;
    FOR(i, 1, N - 1) {
      T a1 = (Y[i + 1] - Y[i]) / (X[i + 1] - X[i]);
      dat.eb(X[i], a1 - a), a = a1;
    }
    return FUNC{ST.new_node(dat), x0, x1, a0, Y[0]};
  }

  pair<T, T> domain(FUNC &f) { return {f.x0, f.x1}; }
  T eval(FUNC &f, T x) {
    auto [x0, x1] = domain(f);
    if (!(x0 <= x && x <= x1)) return infty<T>;
    auto [l, r] = ST.split_max_right(f.root, [&](auto dat) -> bool { return dat.fi <= x; });
    auto [a_sum, xa_sum] = ST.prod(l);
    f.root = ST.merge(l, r);
    return f.y0 + f.a0 * (x - x0) + a_sum * x - xa_sum;
  }
  FUNC restrict_domain(FUNC &f, T L, T R) {
    auto [x0, x1] = domain(f);
    chmax(L, x0), chmin(R, x1);
    if (L > R) {
      ST.free_subtree(f.root), f.root = nullptr;
      f.root = nullptr;
      f.x0 = infty<T>, f.x1 = -infty<T>;
      return f;
    }
    // まずは右側をちぢめる. R 以上の傾き変化を消してしまえばよい
    auto [l, r] = ST.split_max_right(f.root, [&](auto dat) -> bool { return dat.fi < R; });
    ST.free_subtree(r);
    // 左側をちぢめる.
    tie(l, r) = ST.split_max_right(l, [&](auto dat) -> bool { return dat.fi <= L; });
    auto [a_sum, xa_sum] = ST.prod(l);
    T new_a0 = f.a0 + a_sum;
    T new_y0 = f.y0 + f.a0 * (L - x0) + a_sum * L - xa_sum;
    ST.free_subtree(l);
    f.root = r, f.x0 = L, f.x1 = R, f.a0 = new_a0, f.y0 = new_y0;
    return f;
  }
  FUNC add(FUNC &f, FUNC &g) {
    T x0 = max(f.x0, g.x0);
    T x1 = min(f.x1, g.x1);
    restrict_domain(f, x0, x1), restrict_domain(g, x0, x1);
    if (x0 > x1) return f;
    T y0 = f.y0 + g.y0, a0 = f.a0 + g.a0;

    if (len(f) < len(g)) swap(f, g);
    auto tmp = ST.get_all(g.root);
    ST.free_subtree(g.root);
    f.x0 = x0, f.x1 = x1, f.a0 = a0, f.y0 = y0;
    if (!f.root) {
      f.root = ST.new_node(tmp);
      return f;
    }
    // あとは単に tmp を挿入していけばいい
    auto dfs = [&](auto &dfs, np root, int l, int r) -> void {
      if (l == r) return;
      root->prop();
      T x = root->x.fi;
      // [l,m),[m,r)
      int m = binary_search([&](int i) -> bool { return tmp[i].fi >= x; }, r, l - 1, 0);
      if (l < m) {
        if (!root->l) {
          root->l = ST.new_node({tmp.begin() + l, tmp.begin() + m});
        } else {
          dfs(dfs, root->l, l, m);
        }
        root->l->p = root;
      }
      if (m < r) {
        if (!root->r) {
          root->r = ST.new_node({tmp.begin() + m, tmp.begin() + r});
        } else {
          dfs(dfs, root->r, m, r);
        }
        root->r->p = root;
      }
      root->update();
    };
    dfs(dfs, f.root, 0, len(tmp));
    return f;
  }
  FUNC sum_all(vc<FUNC> &funcs) {
    assert(len(funcs) >= 1);
    T x0 = funcs[0].x0, x1 = funcs[0].x1;
    for (auto &g: funcs) chmax(x0, g.x0), chmin(x1, g.x1);
    if (x0 > x1) {
      for (auto &f: funcs) { ST.free_subtree(f.root); }
      return {nullptr, infty<T>, -infty<T>, 0, 0};
    }
    for (auto &f: funcs) f = restrict_domain(f, x0, x1);
    int idx = 0;
    FOR(i, 1, len(funcs)) if (len(funcs[idx]) < len(funcs[i])) idx = i;
    swap(funcs[idx], funcs.back());
    FUNC f = POP(funcs);
    vc<pair<T, T>> dat;
    for (auto &g: funcs) {
      f.y0 += g.y0, f.a0 += g.a0;
      auto tmp = ST.get_all(g.root);
      concat(dat, tmp);
      ST.free_subtree(g.root);
    }
    sort(all(dat));
    // あとは単に dat を挿入していけばいい
    if (!f.root) {
      f.root = ST.new_node(dat);
      return f;
    }
    auto dfs = [&](auto &dfs, np root, int l, int r) -> void {
      if (l == r) return;
      root->prop();
      T x = root->x.fi;
      // [l,m),[m,r)
      int m = binary_search([&](int i) -> bool { return dat[i].fi >= x; }, r, l - 1, 0);
      if (l < m) {
        if (!root->l) {
          root->l = ST.new_node({dat.begin() + l, dat.begin() + m});
        } else {
          dfs(dfs, root->l, l, m);
        }
        root->l->p = root;
      }
      if (m < r) {
        if (!root->r) {
          root->r = ST.new_node({dat.begin() + m, dat.begin() + r});
        } else {
          dfs(dfs, root->r, m, r);
        }
        root->r->p = root;
      }
      root->update();
    };
    dfs(dfs, f.root, 0, len(dat));
    return f;
  }

  FUNC shift(FUNC &f, T add_x, T add_y) {
    ST.apply(f.root, add_x);
    f.x0 += add_x, f.x1 += add_x, f.y0 += add_y;
    return f;
  }

  // h[z]=min(x+y==z)f(x)+g(y)
  FUNC convolve(FUNC &f, FUNC &g) {
    if (f.x0 > f.x1 || g.x0 > g.x1) { return {nullptr, infty<T>, -infty<T>, 0, 0}; }
    if (len(f) < len(g)) swap(f, g);
    shift(f, g.x0, g.y0), shift(g, -g.x0, -g.y0);
    if (len(g) == 0) { return convolve_segment(f, 0, g.x1, g.a0, 0); }
    auto tmp = ST.get_all(g.root);
    ST.free_subtree(g.root);
    f = convolve_segment(f, 0, tmp[0].fi, g.a0, 0);
    T a = g.a0;
    FOR(i, len(tmp)) {
      T nx = (i + 1 < len(tmp) ? tmp[i + 1].fi : g.x1);
      a += tmp[i].se;
      f = convolve_segment(f, 0, nx - tmp[i].fi, a, 0);
      for (auto &[x, a]: ST.get_all(f.root)) {
        assert(f.x0 <= x && x <= f.x1);
        if (f.root) assert(!f.root->p);
      }
    }
    return f;
  }

  // [x0,x1], y=ax+b
  FUNC convolve_segment(FUNC &f, T x0, T x1, T a, T b) {
    assert(x0 <= x1);
    if (f.x0 > f.x1) { return {nullptr, infty<T>, -infty<T>, 0, 0}; }
    f = shift(f, x0, a * x0 + b);
    T d = x1 - x0;
    if (d == 0) return f;
    // (0,0) から (x1,ax1) への線分をどこかに挿入する
    // 特に x0, y0 はこのままでよい
    if (f.x0 == f.x1) { return {nullptr, f.x0, f.x0 + d, a, f.y0}; }
    // 先頭に挿入できる場合
    if (a <= f.a0) {
      ST.apply(f.root, d);
      f.root = ST.merge(ST.new_node({f.x0 + d, f.a0 - a}), f.root);
      f.x1 += d, f.a0 = a;
      return f;
    }
    // 末尾に挿入できる場合
    T a_last = f.a0 + ST.prod(f.root).fi;
    if (a_last <= a) {
      f.root = ST.merge(f.root, ST.new_node({f.x1, a - a_last}));
      f.x1 += d;
      return f;
    }
    // 間のどこかに挿入
    auto [l, r] = ST.split_max_right_prod(f.root, [&](auto prod) -> bool { return f.a0 + prod.fi < a; });
    T asum = ST.prod(l).fi;
    T a1 = a - (asum + f.a0);
    auto [xx, aa] = ST.get(r, 0);
    ST.apply(r, d);
    ST.set(r, 0, {xx + d, aa - a1});
    f.root = ST.merge3(l, ST.new_node({xx, a1}), r);
    f.x1 += d;
    return f;
  }

  FUNC add_const(FUNC &f, T a) {
    f.y0 += a;
    return f;
  }

  FUNC add_linear(FUNC &f, T a, T b) {
    f.y0 += a * f.x0 + b;
    f.a0 += a;
    return f;
  }

  // (a-x)+
  FUNC add_a_minus_x(FUNC &f, T a) {
    auto [x0, x1] = domain(f);
    if (x0 > x1) return f;
    if (a <= x0) return f;
    if (x1 <= a) return add_linear(f, -1, a);
    vc<pair<T, T>> point;
    point.eb(x0, a - x0), point.eb(a, 0), point.eb(x1, 0);
    FUNC g = from_points(point);
    return add(f, g);
  }

  // (x-a)+
  FUNC add_x_minus_a(FUNC &f, T a) {
    auto [x0, x1] = domain(f);
    if (x0 > x1) return f;
    if (a <= x0) return add_linear(f, 1, -a);
    if (x1 <= a) return f;
    vc<pair<T, T>> point;
    point.eb(x0, 0), point.eb(a, 0), point.eb(x1, x1 - a);
    FUNC g = from_points(point);
    return add(f, g);
  }

  // |x-a|
  FUNC add_abs(FUNC &f, T a) {
    f = add_a_minus_x(f, a);
    f = add_x_minus_a(f, a);
    return f;
  }

  // fx,x
  pair<T, T> get_min(FUNC &f) {
    if (f.x0 > f.x1) return {infty<T>, 0};
    if (f.a0 >= 0) { return {f.y0, f.x0}; }
    auto [l, r] = ST.split_max_right_prod(f.root, [&](auto prod) -> bool { return f.a0 + prod.fi < 0; });
    auto [asum, xasum] = ST.prod(l);
    T x = (r ? ST.get(r, 0).fi : f.x1);
    f.root = ST.merge(l, r);
    T y = f.y0 + f.a0 * (x - f.x0) + x * asum - xasum;
    return {y, x};
  }

  FUNC clear_right(FUNC &f) {
    if (f.a0 >= 0) {
      ST.free_subtree(f.root), f.root = nullptr, f.a0 = 0;
      return f;
    }
    auto [l, r] = ST.split_max_right_prod(f.root, [&](auto prod) -> bool { return f.a0 + prod.fi < 0; });
    f.root = l;
    if (!r) { return f; }
    T x = ST.get(r, 0).fi;
    ST.free_subtree(r);
    f.root = ST.merge(f.root, ST.new_node({x, -(f.a0 + ST.prod(l).fi)}));
    return f;
  }
  FUNC clear_left(FUNC &f) {
    if (f.a0 >= 0) { return f; }
    auto [l, r] = ST.split_max_right_prod(f.root, [&](auto prod) -> bool { return f.a0 + prod.fi < 0; });
    auto [asum, xasum] = ST.prod(l);
    if (!r) {
      // 定数にする
      T x = f.x1;
      T y = f.y0 + f.a0 * (x - f.x0) + x * asum - xasum;
      ST.free_subtree(l);
      f.root = nullptr;
      f.y0 = y, f.a0 = 0;
      return f;
    }
    T x = ST.get(f.root, 0).fi;
    T y = f.y0 + f.a0 * (x - f.x0) + x * asum - xasum;
    T a = f.a0 + asum + ST.get(r, 0).se;
    ST.free_subtree(l);
    f.root = r;
    ST.set(r, 0, {x, a});
    f.y0 = y;
    f.a0 = 0;
    return f;
  }
#ifdef FASTIO
  void debug(FUNC &f) {
    auto dat = ST.get_all(f.root);
    SHOW(f.x0, f.x1, f.a0, f.y0);
    SHOW(dat);
  }
#endif
};
} // namespace SLOPE_TRICK_SUPER
using SLOPE_TRICK_SUPER::Slope_Trick_Super;
#line 6 "main.cpp"

void solve() {
  LL(N, M);
  Graph<ll, 1> G(N + M);
  FOR(v, 1, N + M) {
    LL(p, c);
    --p;
    G.add(p, v, c);
  }
  G.build();

  Slope_Trick_Super<i128> ST(2 * (N + M));
  using F = decltype(ST)::FUNC;
  using np = decltype(ST)::np;

  auto dfs = [&](auto& dfs, int v) -> F {
    if (N <= v) { return ST.segment_func(0, infty<ll>, 1, 0); }
    vc<F> funcs;
    funcs.eb(ST.segment_func(0, infty<ll>, 0, 0));
    for (auto& e: G[v]) {
      auto g = dfs(dfs, e.to);
      auto h = ST.segment_func(0, infty<ll>, 0, 0);
      ST.add_abs(h, e.cost);
      g = ST.convolve(g, h);
      g = ST.restrict_domain(g, 0, infty<ll>);
      funcs.eb(g);
    }
    return ST.sum_all(funcs);
  };
  F f = dfs(dfs, 0);
  auto [fx, x] = ST.get_min(f);
  print(fx);
}

signed main() { solve(); }