#line 2 "competitive/std/std.hpp"
#include <bits/stdc++.h>
#ifndef LOCAL_TEST
#pragma GCC target ("avx")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
#endif // LOCAL_TEST
using namespace std;
// 型名の短縮
using ll = long long;
using pii = pair<int, int>; using pll = pair<ll, ll>;
using vi = vector<int>; using vvi = vector<vi>; using vvvi = vector<vvi>;
using vl = vector<ll>; using vvl = vector<vl>; using vvvl = vector<vvl>;
using vb = vector<bool>; using vvb = vector<vb>; using vvvb = vector<vvb>;
using vc = vector<char>; using vvc = vector<vc>; using vvvc = vector<vvc>;
using vd = vector<double>; using vvd = vector<vd>; using vvvd = vector<vvd>;
using vs = vector<string>; using vvs = vector<vector<string>>; using vvvs = vector<vector<vector<string>>>;
template<typename T> vector<vector<T>> vv(int h, int w, T val = T()) { return vector(h, vector<T>(w, val)); }
template<typename T> vector<vector<vector<T>>> vvv(int h1, int h2, int h3, T val = T()) { return vector(h1, vector(h2, vector<T>(h3, val))); }
template<typename T> vector<vector<vector<vector<T>>>> vvvv(int h1, int h2, int h3, int h4, T val = T()) { return vector(h1, vector(h2, vector(h3, vector<T>(h4, val)))); }
template <class T> using priority_queue_min = priority_queue<T, vector<T>, greater<T>>;
// 定数の定義
constexpr double PI = 3.14159265358979323;
constexpr int INF = 100100111; constexpr ll INFL = 3300300300300300491LL;
float EPS = 1e-8; double EPSL = 1e-16;
template<typename T> bool eq(const T x, const T y) { return x == y; }
template<> bool eq<double>(const double x, const double y) { return abs(x - y) < EPSL; }
template<> bool eq<float>(const float x, const float y) { return abs(x - y) < EPS; }
template<typename T> bool neq(const T x, const T y) { return !(eq<T>(x, y)); }
template<typename T> bool ge(const T x, const T y) { return (eq<T>(x, y) || (x > y)); }
template<typename T> bool le(const T x, const T y) { return (eq<T>(x, y) || (x < y)); }
template<typename T> bool gt(const T x, const T y) { return !(le<T>(x, y)); }
template<typename T> bool lt(const T x, const T y) { return !(ge<T>(x, y)); }
constexpr int MODINT998244353 = 998244353;
constexpr int MODINT1000000007 = 1000000007;
// 入出力高速化
struct Nyan { Nyan() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(18); } } nyan;
// 汎用マクロの定義
#define all(a) (a).begin(), (a).end()
#define sz(x) ((ll)(x).size())
#define rep1(n) for(ll dummy_iter = 0LL; dummy_iter < n; ++dummy_iter) // 0 から n-1 まで昇順
#define rep2(i, n) for(ll i = 0LL, i##_counter = 0LL; i##_counter < ll(n); ++(i##_counter), (i) = i##_counter) // 0 から n-1 まで昇順
#define rep3(i, s, t) for(ll i = ll(s), i##_counter = ll(s); i##_counter < ll(t); ++(i##_counter), (i) = (i##_counter)) // s から t まで昇順
#define rep4(i, s, t, step) for(ll i##_counter = step > 0 ? ll(s) : -ll(s), i##_end = step > 0 ? ll(t) : -ll(t), i##_step = abs(step), i = ll(s); i##_counter < i##_end; i##_counter += i##_step, i = step > 0 ? i##_counter : -i##_counter) // s から t まで stepずつ
#define overload4(a, b, c, d, e, ...) e
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)
#define repe(a, v) for(auto& a : (v)) // v の全要素（変更可能）
#define smod(n, m) ((((n) % (m)) + (m)) % (m)) // 非負mod
#define sdiv(n, m) (((n) - smod(n, m)) / (m)) // 非負div
#define uniq(a) {sort(all(a)); (a).erase(unique(all(a)), (a).end());} // 重複除去
int Yes(bool b=true) { cout << (b ? "Yes\n" : "No\n"); return 0; };
int YES(bool b=true) { cout << (b ? "YES\n" : "NO\n"); return 0; };
int No(bool b=true) {return Yes(!b);};
int NO(bool b=true) {return YES(!b);};
template<typename T, size_t N> T max(array<T, N>& a) { return *max_element(all(a)); };
template<typename T, size_t N> T min(array<T, N>& a) { return *min_element(all(a)); };
template<typename T> T max(vector<T>& a) { return *max_element(all(a)); };
template<typename T> T min(vector<T>& a) { return *min_element(all(a)); };
template<typename T> vector<T> vec_slice(const vector<T>& a, int l, int r) { vector<T> rev; rep(i, l, r) rev.push_back(a[i]); return rev; };
template<typename T> T sum(vector<T>& a, T zero = T(0)) { T rev = zero; rep(i, sz(a)) rev += a[i]; return rev; };
template<typename T> bool in_range(const T& val, const T& s, const T& t) { return s <= val && val < t; };
template <class T> inline vector<T>& operator--(vector<T>& v) { repe(x, v) --x; return v; }
template <class T> inline vector<T>& operator++(vector<T>& v) { repe(x, v) ++x; return v; }
// modでのpow
ll powm(ll a, ll n, ll mod=INFL) {
ll res = 1;
while (n > 0) {
if (n & 1) res = (res * a) % mod;
if (n > 1) a = (a * a) % mod;
n >>= 1;
}
return res;
}
// 整数Sqrt
ll sqrtll(ll x) {
assert(x >= 0);
ll rev = sqrt(x);
while(rev * rev > x) --rev;
while((rev+1) * (rev+1)<=x) ++rev;
return rev;
}
template <class T> inline bool chmax(T& M, const T& x) { if (M < x) { M = x; return true; } return false; } // 最大値を更新（更新されたら true を返す）
template <class T> inline bool chmin(T& m, const T& x) { if (m > x) { m = x; return true; } return false; } // 最小値を更新（更新されたら true を返す）
int digit(ll x, int d=10) { int rev=0; while (x > 0) { rev++; x /= d;}; return rev; } // xのd進数桁数
/**
* @brief std.hpp
* @docs docs/std/std.md
*/
#line 3 "competitive/graph/graph.hpp"
template <class Cost> struct Graph{
public:
struct Edge {
int to;
Cost cost;
Edge() {};
Edge(int _to, Cost _cost) : to(_to), cost(_cost) {};
};
struct AdjacencyListRange{
using iterator = typename std::vector<Edge>::const_iterator;
iterator begi, endi;
iterator begin() const { return begi; }
iterator end() const { return endi; }
int size() const { return (int)distance(begi, endi); }
const Edge& operator[](int i) const { return begi[i]; }
};
private:
public:
vector<Edge> E;
vector<int> I;
int n;
Graph() : n(0) {}
Graph(int _n) : n(_n) {}
Graph(int _n, const vector<int>& from, vector<int>& to, vector<Cost>& cost, bool rev = false) : n(_n) {
vector<int> buf(n+1, 0);
for(int i=0; i<(int)from.size(); i++){
++buf[from[i]];
if (rev) ++buf[to[i]];
}
for(int i=1; i<=_n; i++) buf[i] += buf[i-1];
E.resize(buf[n]);
for(int i=(int)from.size()-1; i>=0; i--){
int u = from[i];
int v = to[i];
Cost c = cost[i];
E[--buf[u]] = Edge(v, c);
if(rev) E[--buf[v]] = Edge(u, c);
}
I = move(buf);
}
AdjacencyListRange operator[](int u) const {
return AdjacencyListRange{ E.begin() + I[u], E.begin() + I[u+1] };
}
int num_vertices() const { return n; }
int size() const { return num_vertices(); }
int num_edges() const { return E.size(); }
Graph<Cost> reversed_edges() const {
Graph<Cost> res;
int _n = res.n = n;
vi buf(n+1, 0);
for(auto v : E) ++buf[v.to];
for(int i=1; i<=n; i++) buf[i] += buf[i-1];
res.E.resize(buf[n]);
for(int u=0; u<n; u++) for(auto v : operator[](u)) res.E[--buf[v.to]] = {u, v.cost};
res.I = std::move(buf);
return res;
}
};
template <class T> ostream& operator<<(ostream& os, Graph<T> g) {
bool first = true;
rep(i, g.n) repe(e, g[i]) {
if (first) first = false;
else os << endl;
os << i << "->" << e.to << ": " << e.cost;
}
return os;
}
/**
* @brief graph.hpp
* @docs docs/graph/graph.md
*/
#line 5 "atcoder/segtree.hpp"
#line 2 "atcoder/internal_bit.hpp"
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (unsigned int)(n)) x++;
return x;
}
// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
constexpr int bsf_constexpr(unsigned int n) {
int x = 0;
while (!(n & (1 << x))) x++;
return x;
}
// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
unsigned long index;
_BitScanForward(&index, n);
return index;
#else
return __builtin_ctz(n);
#endif
}
} // namespace internal
} // namespace atcoder
#line 7 "atcoder/segtree.hpp"
namespace atcoder {
template <class S, S (*_op)(S, S), S (*_e)()> struct segtree {
public:
S (*op)(S, S) = _op;
S (*e)() = _e;
segtree() : segtree(0) {}
explicit segtree(const std::vector<S>& v) : _n(int(v.size())) {
log = internal::ceil_pow2(_n);
size = 1 << log;
d = std::vector<S>(2 * size, e());
for (int i = 0; i < _n; i++) d[size + i] = v[i];
for (int i = size - 1; i >= 1; i--) {
update(i);
}
}
explicit segtree(int n) : segtree(std::vector<S>(n, _e())) {}
void set(int p, S x) {
assert(0 <= p && p < _n);
p += size;
d[p] = x;
for (int i = 1; i <= log; i++) update(p >> i);
}
void add(int p, S x) {
assert(0 <= p && p < _n);
(*this).set(p, (*this).get(p) + x);
}
S get(int p) const {
assert(0 <= p && p < _n);
return d[p + size];
}
S prod(int l, int r) const {
assert(0 <= l && l <= r && r <= _n);
S sml = e(), smr = e();
l += size;
r += size;
while (l < r) {
if (l & 1) sml = op(sml, d[l++]);
if (r & 1) smr = op(d[--r], smr);
l >>= 1;
r >>= 1;
}
return op(sml, smr);
}
S all_prod() const { return d[1]; }
template <bool (*f)(S)> int max_right(int l) const {
return max_right(l, [](S x) { return f(x); });
}
template <class F> int max_right(int l, F f) const {
assert(0 <= l && l <= _n);
assert(f(e()));
if (l == _n) return _n;
l += size;
S sm = e();
do {
while (l % 2 == 0) l >>= 1;
if (!f(op(sm, d[l]))) {
while (l < size) {
l = (2 * l);
if (f(op(sm, d[l]))) {
sm = op(sm, d[l]);
l++;
}
}
return l - size;
}
sm = op(sm, d[l]);
l++;
} while ((l & -l) != l);
return _n;
}
template <bool (*f)(S)> int min_left(int r) const {
return min_left(r, [](S x) { return f(x); });
}
template <class F> int min_left(int r, F f) const {
assert(0 <= r && r <= _n);
assert(f(e()));
if (r == 0) return 0;
r += size;
S sm = e();
do {
r--;
while (r > 1 && (r % 2)) r >>= 1;
if (!f(op(d[r], sm))) {
while (r < size) {
r = (2 * r + 1);
if (f(op(d[r], sm))) {
sm = op(d[r], sm);
r--;
}
}
return r + 1 - size;
}
sm = op(d[r], sm);
} while ((r & -r) != r);
return 0;
}
int n() const {return (*this)._n;}
private:
int _n, size, log;
std::vector<S> d;
void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
};
} // namespace atcoder
#line 4 "competitive/data_structure/segtree.hpp"
template <typename S, S (*op)(S, S), S (*e)()> std::ostream& operator<<(std::ostream& os, const atcoder::segtree<S, op, e> seg) {
int n = seg.n();
rep(i, n) { os << seg.get(i); if (i != n-1) os << " "; }
return os;
};
namespace segtree_internal {
template<typename T> T op_max(T x, T y) { return x > y? x : y; }
template<typename T> T op_min(T x, T y) { return x < y? x : y; }
template<typename T> T op_add(T x, T y) { return x + y; }
template<typename T> T e_max() { return -INFL; }
template<> int e_max() { return -INF; }
template<typename T> T e_min() { return INFL; }
template<> int e_min() { return INF; }
template<typename T> T e_add() { return 0; }
}
template<class S, S (*op)(S,S), S(*e)()> using segtree = atcoder::segtree<S, op, e>;
template<typename T> using seg_add = segtree<T, segtree_internal::op_add<T>, segtree_internal::e_add<T>>;
template<typename T> using seg_max = segtree<T, segtree_internal::op_max<T>, segtree_internal::e_max<T>>;
template<typename T> using seg_min = segtree<T, segtree_internal::op_min<T>, segtree_internal::e_min<T>>;
/**
* @brief セグメント木（ラッパー）
* @docs docs/data_structure/segtree.md
*/
#line 6 "atcoder/lazysegtree.hpp"
#line 8 "atcoder/lazysegtree.hpp"
namespace atcoder {
template <class S,
S (*_op)(S, S),
S (*_e)(),
class F,
S (*_mapping)(F, S),
F (*_composition)(F, F),
F (*_id)()>
struct lazy_segtree {
public:
S (*op)(S, S) = _op;
S (*e)() = _e;
S (*mapping)(F, S) = _mapping;
F (*composition)(F, F) = _composition;
F (*id)() = _id;
lazy_segtree() : lazy_segtree(0) {}
explicit lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, _e())) {}
explicit lazy_segtree(const std::vector<S>& v) : _n(int(v.size())) {
log = internal::ceil_pow2(_n);
size = 1 << log;
d = std::vector<S>(2 * size, e());
lz = std::vector<F>(size, id());
for (int i = 0; i < _n; i++) d[size + i] = v[i];
for (int i = size - 1; i >= 1; i--) {
update(i);
}
}
void set(int p, S x) {
assert(0 <= p && p < _n);
p += size;
for (int i = log; i >= 1; i--) push(p >> i);
d[p] = x;
for (int i = 1; i <= log; i++) update(p >> i);
}
void add(int p, S x) {
assert(0 <= p && p < _n);
(*this).set(p, (*this).get(p) + x);
}
S get(int p) {
assert(0 <= p && p < _n);
p += size;
for (int i = log; i >= 1; i--) push(p >> i);
return d[p];
}
S prod(int l, int r) {
assert(0 <= l && l <= r && r <= _n);
if (l == r) return e();
l += size;
r += size;
for (int i = log; i >= 1; i--) {
if (((l >> i) << i) != l) push(l >> i);
if (((r >> i) << i) != r) push((r - 1) >> i);
}
S sml = e(), smr = e();
while (l < r) {
if (l & 1) sml = op(sml, d[l++]);
if (r & 1) smr = op(d[--r], smr);
l >>= 1;
r >>= 1;
}
return op(sml, smr);
}
S all_prod() { return d[1]; }
void apply(int p, F f) {
assert(0 <= p && p < _n);
p += size;
for (int i = log; i >= 1; i--) push(p >> i);
d[p] = mapping(f, d[p]);
for (int i = 1; i <= log; i++) update(p >> i);
}
void apply(int l, int r, F f) {
assert(0 <= l && l <= r && r <= _n);
if (l == r) return;
l += size;
r += size;
for (int i = log; i >= 1; i--) {
if (((l >> i) << i) != l) push(l >> i);
if (((r >> i) << i) != r) push((r - 1) >> i);
}
{
int l2 = l, r2 = r;
while (l < r) {
if (l & 1) all_apply(l++, f);
if (r & 1) all_apply(--r, f);
l >>= 1;
r >>= 1;
}
l = l2;
r = r2;
}
for (int i = 1; i <= log; i++) {
if (((l >> i) << i) != l) update(l >> i);
if (((r >> i) << i) != r) update((r - 1) >> i);
}
}
template <bool (*g)(S)> int max_right(int l) {
return max_right(l, [](S x) { return g(x); });
}
template <class G> int max_right(int l, G g) {
assert(0 <= l && l <= _n);
assert(g(e()));
if (l == _n) return _n;
l += size;
for (int i = log; i >= 1; i--) push(l >> i);
S sm = e();
do {
while (l % 2 == 0) l >>= 1;
if (!g(op(sm, d[l]))) {
while (l < size) {
push(l);
l = (2 * l);
if (g(op(sm, d[l]))) {
sm = op(sm, d[l]);
l++;
}
}
return l - size;
}
sm = op(sm, d[l]);
l++;
} while ((l & -l) != l);
return _n;
}
template <bool (*g)(S)> int min_left(int r) {
return min_left(r, [](S x) { return g(x); });
}
template <class G> int min_left(int r, G g) {
assert(0 <= r && r <= _n);
assert(g(e()));
if (r == 0) return 0;
r += size;
for (int i = log; i >= 1; i--) push((r - 1) >> i);
S sm = e();
do {
r--;
while (r > 1 && (r % 2)) r >>= 1;
if (!g(op(d[r], sm))) {
while (r < size) {
push(r);
r = (2 * r + 1);
if (g(op(d[r], sm))) {
sm = op(d[r], sm);
r--;
}
}
return r + 1 - size;
}
sm = op(d[r], sm);
} while ((r & -r) != r);
return 0;
}
int n() {return (*this)._n;}
private:
int _n, size, log;
std::vector<S> d;
std::vector<F> lz;
void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
void all_apply(int k, F f) {
d[k] = mapping(f, d[k]);
if (k < size) lz[k] = composition(f, lz[k]);
}
void push(int k) {
all_apply(2 * k, lz[k]);
all_apply(2 * k + 1, lz[k]);
lz[k] = id();
}
};
} // namespace atcoder
#line 4 "competitive/data_structure/lazysegtree.hpp"
template <typename S, S (*op)(S, S), S (*e)(), class F, S (*mapping)(F, S), F (*composition)(F, F), F (*id)()>
std::ostream& operator<<(std::ostream& os, atcoder::lazy_segtree<S, op, e, F, mapping, composition, id> seg) {
int n = seg.n();
rep(i, n) { os << seg.get(i); if (i != n-1) os << " "; }
return os;
};
namespace lsegtree_internal {
template<typename T> struct AddNode {
T value;
ll size;
AddNode() : value(T(0)), size(1) {};
AddNode(T value, ll size) : value(value), size(size) {};
friend ostream& operator<<(std::ostream& os, const AddNode<T> &n) { os << n.value; return os; };
};
template<typename T> T e_max() { return -INFL; }
template<> int e_max() { return -INF; }
template<typename T> T e_min() { return INFL; }
template<> int e_min() { return INF; }
template<typename T> AddNode<T> e_add() { return {0, 1}; }
template<typename T> T op_max(T x, T y) { return x > y ? x : y; }
template<typename T> T op_min(T x, T y) { return x < y ? x : y; }
template<typename T> AddNode<T> op_add(AddNode<T> x, AddNode<T> y) { return {x.value + y.value, x.size + y.size}; }
template<typename T> T id_radd(){ return 0; }
template<typename T> T id_rupdate(){ return INFL; }
template<> int id_rupdate(){ return INF; }
template<typename T> AddNode<T> mapping_add_radd(T f, AddNode<T> x){ return {x.value + f * x.size, x.size}; }
template<typename T> AddNode<T> mapping_add_rupdate(T f, AddNode<T> x){
AddNode<T> rev = AddNode<T>(x);
if(f != id_rupdate<T>()) rev.value = f * rev.size;
return rev;
}
template<typename T> T mapping_radd(T f, T x){ return f+x; }
template<typename T> T mapping_rupdate(T f, T x){ return (f == id_rupdate<T>() ? x : f); }
template<typename T> T composition_radd(T f, T g){ return f+g; }
template<typename T> T composition_rupdate(T f, T g){ return (f == id_rupdate<T>() ? g : f); }
}
template <typename S, S (*op)(S, S), S (*e)(), class F, S (*mapping)(F, S), F (*composition)(F, F), F (*id)()>
using lsegtree = atcoder::lazy_segtree<S, op, e, F, mapping, composition, id>;
template<typename T> using lseg_add_radd = atcoder::lazy_segtree<lsegtree_internal::AddNode<T>, lsegtree_internal::op_add<T>, lsegtree_internal::e_add<T>, T, lsegtree_internal::mapping_add_radd<T>, lsegtree_internal::composition_radd<T>, lsegtree_internal::id_radd<T>>;
template<typename T> using lseg_min_radd = atcoder::lazy_segtree<T, lsegtree_internal::op_min<T>, lsegtree_internal::e_min<T>, T, lsegtree_internal::mapping_radd<T>, lsegtree_internal::composition_radd<T>, lsegtree_internal::id_radd<T>>;
template<typename T> using lseg_max_radd = atcoder::lazy_segtree<T, lsegtree_internal::op_max<T>, lsegtree_internal::e_max<T>, T, lsegtree_internal::mapping_radd<T>, lsegtree_internal::composition_radd<T>, lsegtree_internal::id_radd<T>>;
template<typename T> using lseg_add_rupdate = atcoder::lazy_segtree<lsegtree_internal::AddNode<T>, lsegtree_internal::op_add<T>, lsegtree_internal::e_add<T>, T, lsegtree_internal::mapping_add_rupdate<T>, lsegtree_internal::composition_rupdate<T>, lsegtree_internal::id_rupdate<T>>;
template<typename T> using lseg_min_rupdate = atcoder::lazy_segtree<T, lsegtree_internal::op_min<T>, lsegtree_internal::e_min<T>, T, lsegtree_internal::mapping_rupdate<T>, lsegtree_internal::composition_rupdate<T>, lsegtree_internal::id_rupdate<T>>;
template<typename T> using lseg_max_rupdate = atcoder::lazy_segtree<T, lsegtree_internal::op_max<T>, lsegtree_internal::e_max<T>, T, lsegtree_internal::mapping_rupdate<T>, lsegtree_internal::composition_rupdate<T>, lsegtree_internal::id_rupdate<T>>;
/**
* @brief 遅延セグメント木（ラッパー）
* @docs docs/data_structure/lazysegtree.md
*/
#line 6 "competitive/tree/heavy_light_decomposition.hpp"
template <typename Cost, typename Seg> struct HeavyLightDecomposition {
vi heavy_edge,in,out,head,par,pos;
Seg &seg;
bool edge;
HeavyLightDecomposition(Graph<Cost>& g, Seg& seg, bool edge = true) :
heavy_edge(g.n), in(g.n), out(g.n), head(g.n), par(g.n), pos(g.n), seg(seg), edge(edge) {
build(g);
}
void build(Graph<Cost>& g) {
vi subtree_size(g.n, 0);
auto dfs_sz = [&] (auto self, int x, int p) -> int {
par[x] = p;
subtree_size[x] = 1;
repe(e, g[x]) {
if (e.to == p) continue;
subtree_size[x] += self(self, e.to, x);
}
int maxs = -INF;
heavy_edge[x] = -1;
repe(e, g[x]) {
if (e.to == p) continue;
if (chmax(maxs, subtree_size[e.to])) heavy_edge[x] = e.to;
}
return subtree_size[x];
};
dfs_sz(dfs_sz, 0, -1);
int t = 0;
auto dfs_hld = [&] (auto self, int x, int par) -> void {
in[x] = t++;
pos[in[x]] = x;
if (heavy_edge[x] != -1) {
head[heavy_edge[x]] = head[x];
self(self, heavy_edge[x], x);
}
repe(e, g[x]) {
if (e.to == par || e.to == heavy_edge[x]) continue;
head[e.to] = e.to;
self(self, e.to, x);
}
out[x] = t;
};
dfs_hld(dfs_hld, 0, -1);
}
int lca(int u, int v) {
while (head[u] != head[v]) {
if (in[u] > in[v]) swap(u, v);
v = par[head[v]];
}
return in[u] < in[v] ? u : v;
}
int la(int u, int k) {
while (true) {
if (u == -1) return -1;
if (u == 0 && k > 0) return -1;
int v = head[u];
if (in[u] - k >= in[v]) return pos[in[u] - k];
k -= in[u] - in[v] + 1;
u = par[head[u]];
}
}
decltype(seg.e()) prod(int u, int v) {
using T = decltype(seg.e());
T l = seg.e();
T r = seg.e();
while (head[u] != head[v]) {
if (in[u] > in[v]) swap(u, v), swap(l, r);
l = seg.op(seg.prod(in[head[v]], in[v] + 1), l);
v = par[head[v]];
}
if (in[u] > in[v]) swap(u, v), swap(l, r);
// パスクエリの場合はu（uからuの親へのパス）は足さない
return seg.op(seg.op(seg.prod(in[u] + edge, in[v] + 1), l) , r);
}
template<typename F> void apply(int u, int v, F f) {
while (head[u] != head[v]) {
if (in[u] > in[v]) swap(u, v);
seg.apply(in[head[v]], in[v] + 1, f);
v = par[head[v]];
}
if (in[u] > in[v]) swap(u, v);
seg.apply(in[u] + edge, in[v] + 1, f);
}
int edge_pos(int u, int v) {
if (par[u] != v) swap(u, v);
assert(par[u] == v);
return in[u];
}
};
/**
* @brief HL分解 (Heavy Light Decomposition)
* @docs docs/tree/heavy_light_decomposition.md
*/
HL分解 (Heavy Light Decomposition)