#pragma warning(disable:4996)
#include <bits/stdc++.h> 
using namespace std;
#if __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
using mint = static_modint<998244353>;
using vmint = vector<mint>;
using vvmint = vector<vmint>;
using vvvmint = vector<vvmint>;
using vvvvmint = vector<vvvmint>;
istream& operator>>(istream& is, mint& a) {
	long long v;
	is >> v;
	a = v;
	return is;
}
ostream& operator<<(ostream& os, mint a) {
	return os << a.val();
}
#endif
#ifdef _MSC_VER
#include <intrin.h>
#define __builtin_popcount __popcnt
#define __builtin_popcountll __popcnt64
#endif
template<typename T>
using vec = vector<T>;
using ll = long long;
using ull = unsigned long long;
using stirng = string;
using itn = int;
using vs = vec<string>;
using vc = vec<char>;
using vvc = vec<vc>;
using vvvc = vec<vvc>;
using vi = vec<int>;
using vvi = vec<vi>;
using vvvi = vec<vvi>;
using vvvvi = vec<vvvi>;
using vll = vec<ll>;
using vvll = vec<vll>;
using vvvll = vec<vvll>;
using vvvvll = vec<vvvll>;
using si = set<int>;
using vsi = vec<si>;
using vvsi = vec<vsi>;
using vvvsi = vec<vvsi>;
using sll = set<ll>;
using vsll = vec<sll>;
using vvsll = vec<vsll>;
using vvvsll = vec<vvsll>;
using vd = vec<double>;
using vvd = vec<vd>;
using vvvd = vec<vvd>;
using vvvvd = vec<vvvd>;
using pii = pair<int, int>;
using pll = pair<ll, ll>;
using pd = pair<double, double>;
using mii = map<int, int>;
using mll = map<ll, ll>;
using msi = multiset<int>;
using msll = multiset<ll>;
using vpii = vec<pii>;
using vvpii = vec<vpii>;
using vvvpii = vec<vvpii>;
using vpll = vec<pll>;
using vvpll = vec<vpll>;
using vvvpll = vec<vvpll>;
using vpd = vec<pd>;
using spii = set<pii>;
using spll = set<pll>;
using spd = set<pd>;
using vspll = vec<spll>;
using vvspll = vec<vspll>;
using vb = vec<bool>;
using vvb = vec<vb>;
using vvvb = vec<vvb>;
#define rep(i,j,n) for(ll i=j;i<(n);i++)
#define rrep(i,j,n) for (ll i = (n)-1; i >=j; i--)
#define Call_4th(_1, _2, _3, _4, ...) _4
#define rp(...) Call_4th(__VA_ARGS__, _1, rp2)(__VA_ARGS__)
#define rp2(i, n) rep(i, 0, n)
#define rrp(i,n) rrep(i,0,(n))
#define all(v) v.begin(), v.end()
#define pb push_back
#define pp pop_back()
#define bk back()
#define ins insert
#define yes cout<<"Yes"<<endl;
#define no cout<<"No"<<endl; 
#define NL {cout<<endl;}
#define n0 {cout<<"-1"<<endl;}
#define lwb lower_bound
#define upb upper_bound
#define reutrn return
#define Endl endl
template<typename T>
int len(T& x) { return int(x.size()); }
#define sz size()
#define fi first
#define se second
#define nxp next_permutation
ll mod0 = 998244353, lmax = 4e18, inf = 1050000000;
double pi = 3.141592653589793238462643;

//cin系統
template<typename... T>
void in(T&... X) { (cin >> ... >> X); }
template<typename T>
void ain(vec<T>& Y) { rp(K, Y.size()) cin >> Y[K]; }
template<typename T>
void ain(set<T>& Y, auto n) {
	rp(i, n) {
		T YY; cin >> YY;
		Y.insert(YY);
	}
}
template<typename T>
void ain(multiset<T>& Y, auto n) {
	rp(i, n) {
		T YY; cin >> YY;
		Y.insert(YY);
	}
}
template<typename T>
void ain(pair<T, T>& a) { cin >> a.fi >> a.se; }
template<typename T>
void bin(vec<T>& Z) { rp(L, Z.size()) ain(Z[L]); }

//cout系統
// vector
template <typename T> struct is_vector : false_type {};
template <typename T, typename Alloc>
struct is_vector<vector<T, Alloc>> : true_type {};
// queue
template <typename T> struct is_queue : false_type {};
template <typename T, typename Container>
struct is_queue<queue<T, Container>> : true_type {};
// stack
template <typename T> struct is_stack : false_type {};
template <typename T, typename Container>
struct is_stack<stack<T, Container>> : true_type {};
// pair
template <typename T> struct is_pair : false_type {};
template <typename T1, typename T2>
struct is_pair<pair<T1, T2>> : true_type {};
// set / multiset
template <typename T> struct is_set_like : false_type {};
template <typename Key, typename Compare, typename Alloc>
struct is_set_like<set<Key, Compare, Alloc>> : true_type {};
template <typename Key, typename Compare, typename Alloc>
struct is_set_like<multiset<Key, Compare, Alloc>> : true_type {};
// map / multimap
template <typename T> struct is_map_like : false_type {};
template <typename Key, typename Val, typename Compare, typename Alloc>
struct is_map_like<map<Key, Val, Compare, Alloc>> : true_type {};
template <typename Key, typename Val, typename Compare, typename Alloc>
struct is_map_like<multimap<Key, Val, Compare, Alloc>> : true_type {};

template <typename T>
void aout(const T& x) {
	if constexpr (is_vector<T>::value) {
		for (const auto& e : x) aout(e);
		cout << endl;
	}
	else if constexpr (is_queue<T>::value) {
		auto copy = x;
		while (!copy.empty()) {
			aout(copy.front());
			copy.pop();
		}
		cout << endl;
	}
	else if constexpr (is_stack<T>::value) {
		auto copy = x;
		while (!copy.empty()) {
			aout(copy.top());
			copy.pop();
		}
		cout << endl;
	}
	else if constexpr (is_pair<T>::value) {
		aout(x.first);
		aout(x.second);
		cout << endl;
	}
	else if constexpr (is_set_like<T>::value) {
		for (const auto& e : x) aout(e);
		cout << endl;
	}
	else if constexpr (is_map_like<T>::value) {
		for (const auto& e : x) aout(e); // pair<const Key, Val>
		cout << endl;
	}
	else {
		cout << x << " ";
	}
}

//数学系
ll modcnt(ll l, ll r, ll m, ll a) {//l以上r以下の整数iでi≡a(mod m)となるものの個数(l>r,a<0等も可)
	assert(m != 0);
	m = abs(m);
	ll x = l + ((a - l) % m + m) % m;//l以上で最小のi≡a(mod m)となるi
	if (x > r) return 0;
	return (r - x) / m + 1;
}
ll dsum(ll a) {//桁和
	ll r = 0;
	while (a != 0) {
		r += a % 10;
		a /= 10;
	}
	return r;
}
ll dnum(ll a) {//桁数(0は0桁)
	ll r = 0;
	while (a != 0) {
		r++;
		a /= 10;
	}
	return r;
}
sll yk(ll n) {//約数全列挙O(sqrt(N)),set<int>
	sll yaku;
	for (ll i = 1; i * i <= n; i++) if (n % i == 0) {
		yaku.insert(i);
		yaku.insert(n / i);
	}
	return yaku;
}
vi ssa(int n) {//n:必要な素数の最大値 素数全列挙 線形篩
	vector<int> prime;
	vector<int>lpf(n + 1, 0);//最小素因数
	rep(i, 2, n + 1) {
		if (lpf[i] == 0) {
			lpf[i] = i;
			prime.push_back(i);
		}
		for (int k : prime) {
			if (k * i > n || k > lpf[i]) break;//k最小素因数よりk<=i
			lpf[k * i] = k;//kを最小素因数に持つ数k*iを拾う
		}
	}
	return prime;
}
vector<pll> sib(ll x) {//素因数分解
	if (x == 1) return vector<pll>(1, { 1,1 });
	vector<pll> ans;
	for (ll i = 2; i * i <= x; i++) {//rootx回iを回しxで割る
		ll kaisu = 0;
		while (x % i == 0) {
			x /= i; kaisu++;
		}
		if (kaisu > 0) ans.pb({ i,kaisu });
	}
	if (x != 1) ans.pb({ x,1 });
	return ans;
}
bool isp(ll x) {//素数判定 O(sqrt(x))
	auto y = sib(x);
	return (x != 1 && y.sz == 1 && y[0] == pll{ x, 1 });
}
vec<vec<pll>> sib(ll x, ll y) {//閉区間[x,y]全素因数分解 x=1は何も返さない O(max(sqrt(y),y-x)loglogy) ABC227G ABC412E
	vi b = ssa(sqrt(y) + 100);
	vec<vec<pll>> r(y - x + 1);
	vll num(y - x + 1);
	rp(i, y - x + 1) num[i] = x + i;
	for (ll k : b) for (ll a = (1 + (x - 1) / k) * k; a <= y; a += k) {
		ll kaisu = 0;
		while (num[a - x] % k == 0) {
			num[a - x] /= k; kaisu++;
		}
		if (kaisu) r[a - x].pb({ k,kaisu });
	}
	rp(i, y - x + 1) if (num[i] != 1) r[i].pb({ num[i],1 });
	return r;
}
template<typename T>
ll sqrtll(T a) {//整数sqrt a<0orr非整数の場合-1を返す
	if (a < 0) return -1;
	ll r = sqrt(a);
	while (r * r > a) r--;
	while ((r + 1) * (r + 1) <= a) r++;
	if (r * r == a)return r;
	else return -1;
}
#define llsqrt sqrtll
ll powll(ll a, ll b) {//llでpow
	ll c = 1;
	assert(pow(a, b) <= 9e18);
	rp(i, b) {
		c *= a;
	}
	return c;
}
#define llpow powll
template<typename T>
T sq(T a) {//2乗
	return a * a;
}
template<typename T>
T cb(T a) {//3乗
	return a * a * a;
}
string basea(ll x, ll a) {//xをa進法に変換
	string r;
	while (x != 0) {
		r.pb(x % a + '0');
		x /= a;
	}
	reverse(all(r));
	return r;
}
ll base10(string x, ll a) {//a進数のxを10進法に戻す
	ll r = 0;
	rp(i, x.sz) {
		r *= a;
		r += x[i] - '0';
	}
	return r;
}
vll vbasea(ll x, ll a) {//xをa進法に変換(vector)
	vll r;
	while (x != 0) {
		r.pb(x % a);
		x /= a;
	}
	reverse(all(r));
	return r;
}
template<typename T>
ll bc(T x) {//bitcount
	assert(x >= 0);
	return __builtin_popcount(x);
}
double ds(double x1, double x2, double y1, double y2) {//dist
	return sqrt((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2));
}
ll ds2(ll x1, ll x2, ll y1, ll y2) {//distの2乗(整数値)
	return (x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2);
}
template<typename T>
ll ds2(vec<T> a) {
	return (a[0] - a[2]) * (a[0] - a[2]) + (a[1] - a[3]) * (a[1] - a[3]);
}
ll gpsum(ll a, ll r, ll n, ll m) {//初項a,公比r,項数nの等比数列の総和mod m r=10でレピュニットも可能
	if (n == 0) return 0;
	if (n == 1) return a % m;
	modint::set_mod(m);
	if (n % 2 == 1) return (a + gpsum(a * r % m, r, n - 1, m)) % m;
	else return ((modint(r).pow(n / 2) + 1) * gpsum(a, r, n / 2, m)).val();
}

//二分探索
template<typename T>
auto lwa(typename std::vector<T>::iterator b, typename std::vector<T>::iterator e, T c) {//[b,e)でcより小さい最大の値のイテレータ(複数あれば右端)
	auto it = std::lower_bound(b, e, c);
	if (it == b) return e;
	--it;
	return it;
}
template<typename T>
auto lwa(vector<T>& a, T b) {//aでbより小さい最大の値のイテレータ(複数あれば右端)
	return lwa(all(a), b);
}
template<typename T>
auto lwa(set<T>& a, T b) {//aでbより小さい最大の値のイテレータ(複数あれば右端)
	auto i = a.lwb(b);
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}
template<typename T>
auto lwa(multiset<T>& a, T b) {//aでbより小さい最大の値のイテレータ(複数あれば右端)
	auto i = a.lwb(b);
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}

template<typename T>
auto lwc(typename std::vector<T>::iterator b, typename std::vector<T>::iterator e, T c) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	auto i = lwb(b, e, c);
	if (i != e && *i == c) return i;
	if (i == b) return e;
	else i--;
	return i;
}
template<typename T>
auto lwc(vector<T>& a, T b) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	return lwc(all(a), b);
}
template<typename T>
auto lwc(set<T>& a, T b) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	auto i = a.lwb(b);
	if (i != a.end() && *i == b) return i;
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}
template<typename T>
auto lwc(multiset<T>& a, T b) {//aでb以下の最大の値のイテレータ(複数あれば左端)
	auto i = a.lwb(b);
	if (i != a.end() && *i == b) return i;
	if (i == a.begin()) return a.end();
	else i--;
	return i;
}

template <typename T>
std::pair<typename vector<T>::iterator, typename vector<T>::iterator> its(vec<T>& v, T a) {//{--lwb,lwb}({x未満の右端,x以上の左端})のpair返す(なければend())
	auto it = lwb(all(v), a);
	if (it == v.begin()) return { v.end(), it };
	auto it2 = it; it2--;
	return { it2,it };
}
template <typename T>
std::pair<typename set<T>::iterator, typename set<T>::iterator> its(set<T>& v, T a) {//{--lwb,lwb}({x未満の右端,x以上の左端})のpair返す(なければend())
	auto it = v.lwb(a);
	if (it == v.begin()) return { v.end(), it };
	auto it2 = it; it2--;
	return { it2,it };
}
template <typename T>
std::pair<typename multiset<T>::iterator, typename multiset<T>::iterator> its(multiset<T>& v, T a) {//{--lwb,lwb}({x未満の右端,x以上の左端})のpair返す(なければend())
	auto it = v.lwb(a);
	if (it == v.begin()) return { v.end(), it };
	auto it2 = it; it2--;
	return { it2,it };
}

template <typename T>
typename vector<T>::iterator nits(vec<T>& v, T a, bool isr = 1) {//itsのうちaに近い方返す(距離同じの場合isr==0なら左優先,1なら右優先) v.sz==0のときv.end()返す
	auto cd = its(v, a);
	if (cd.fi == v.end()) return cd.se;
	if (cd.se == v.end()) return cd.fi;
	auto l = a - *cd.fi, r = *cd.se - a;
	if (l < r) return cd.fi;
	if (l > r || isr == 1) return cd.se;
	return cd.fi;
}
template <typename T>
typename set<T>::iterator nits(set<T>& v, T a, bool isr = 1) {//itsのうちaに近い方返す(距離同じの場合isr==0なら左優先,1なら右優先) v.sz==0のときv.end()返す
	auto cd = its(v, a);
	if (cd.fi == v.end()) return cd.se;
	if (cd.se == v.end()) return cd.fi;
	auto l = a - *cd.fi, r = *cd.se - a;
	if (l < r) return cd.fi;
	if (l > r || isr == 1) return cd.se;
	return cd.fi;
}
template <typename T>
typename multiset<T>::iterator nits(multiset<T>& v, T a, bool isr = 1) {//itsのうちaに近い方返す(距離同じの場合isr==0なら左優先,1なら右優先) v.sz==0のときv.end()返す
	auto cd = its(v, a);
	if (cd.fi == v.end()) return cd.se;
	if (cd.se == v.end()) return cd.fi;
	auto l = a - *cd.fi, r = *cd.se - a;
	if (l < r) return cd.fi;
	if (l > r || isr == 1) return cd.se;
	return cd.fi;
}

template <typename T>
int rcount(vec<T>& a, T l, T r) {//ソート済配列aから[l,r]の範囲(l以上r以下)の要素カウント
	//右端を超える最小のi-左端に引っかかる最小のi
	//l,rのどちらに対しても境界を含む/含まないを入れ替えるときupb/lwbを変えれば良い
	return upb(all(a), r) - lwb(all(a), l);
}
template <typename T>
int rcount(set<T>& a, T l, T r) {//ソート済配列aから[l,r]の範囲(l以上r以下)の要素カウント
	return a.upb(r) - a.lwb(l);
}

template <class Container, class Key>
bool has(const Container& c, const Key& x) {//sort済vector/set/map等に対しその値を持つかどうか返す
	if constexpr (is_set_like<Container>::value || is_map_like<Container>::value) {
		// set / multiset / map
		return c.find(x) != c.end();
	}
	else {
		// vector 等（ソート済前提）
		return std::binary_search(c.begin(), c.end(), x);
	}
}

//配列便利関数系
template<typename T>
bool chmin(T& M1, T M2) {
	if (M1 > M2) { M1 = M2; return 1; }
	else return 0;
}
template<typename T>
bool chmax(T& M1, T M2) {
	if (M1 < M2) { M1 = M2; return 1; }
	else return 0;
}
template<typename T>
auto vmax(T a) {
	return *max_element(all(a));
}
template<typename T>
auto vmin(T a) {
	return *min_element(all(a));
}
template<typename T>
auto vmaxi(T a) {
	return max_element(all(a)) - a.begin();
}
template<typename T>
auto vmini(T a) {
	return min_element(all(a)) - a.begin();
}
template<typename T>
auto vsum(vector<T> a) {
	T x = 0;
	for (auto i : a)x += i;
	return x;
}
template<typename T>
vec<T> inv(vec<T> a) {//0-indexed 逆関数
	vec<T> ans(a.sz);
	rp(i, a.sz) {
		ans[a[i]] = i;
	}
	return ans;
}
template<typename T>
vec<T> inv1(vec<T> a) {//1-indexed
	vec<T> ans(a.sz);
	rp(i, a.sz) {
		ans[a[i] - 1] = i + 1;
	}
	return ans;
}
template<typename T>
vec<T> zat(vec<T> a) {//座標圧縮 T=ll,int
	auto b = a;
	sort(all(b));
	b.erase(unique(all(b)), b.end());
	vec<T> c(a.size());
	rp(i, a.size()) {
		c[i] = lwb(all(b), a[i]) - b.begin();
	}
	return c;
}
template<typename T>
vec<T> rsw(vec<T> a, bool is_put0 = 1) {//累積和 T=ll,int
	if (is_put0 == 0) {
		vec<T> b = {};
		rp(i, a.sz) b.pb(a[i] + (b.sz == 0 ? 0 : b[b.sz - 1]));
		return b;
	}
	vec<T> b = { 0 };
	rp(i, a.sz) b.pb(a[i] + b[b.sz - 1]);
	return b;
}
template<typename T>
T rswsum(vec<T>& a, int l, int r, bool is_put0 = 1) {//rsw配列から元の配列の[l,r)番目(0indexed)の和を取得
	if (is_put0 == 0) {
		assert(0 <= l && l <= r);
		assert(r <= a.sz);
		return (r == 0 ? 0 : a[r - 1]) - (l == 0 ? 0 : a[l - 1]);
	}
	assert(0 <= l && l <= r);
	assert(r < a.sz);
	return a[r] - a[l];
}
template<typename T>
T rswsum2(vec<T>& a, int l, int r, bool is_put0 = 1) {//rsw配列から元の配列の[l,r]番目(0indexed)の和を取得
	return rswsum(a, l, r + 1, is_put0);
}

//文字列系
vector<pair<char, ll>>  rle(string rr) {
	vector<pair<char, ll>> ret;
	char lc;
	rep(i, 0, rr.size()) {
		if (i > 0 && rr[i] == lc) ret[ret.sz - 1].second++;
		else { ret.pb({ rr[i],1 }); }
		lc = rr[i];
	}
	return ret;
}
vector<pair<int, ll>>  rle(vi rr) {
	vector<pair<int, ll>> ret;
	int lc;
	rep(i, 0, rr.size()) {
		if (i > 0 && rr[i] == lc) ret[ret.sz - 1].second++;
		else { ret.pb({ rr[i],1 }); }
		lc = rr[i];
	}
	return ret;
}
vector<pair<ll, ll>>  rle(vll rr) {
	vector<pair<ll, ll>> ret;
	ll lc;
	rep(i, 0, rr.size()) {
		if (i > 0 && rr[i] == lc) ret[ret.sz - 1].second++;
		else { ret.pb({ rr[i],1 }); }
		lc = rr[i];
	}
	return ret;
}
bool kai(string S) {//回文か判定
	bool ans = 1;
	rp(i, S.size()) if (S[i] != S[S.size() - 1 - i]) ans = 0;
	return ans;
}
template <typename... Args>
string concat(const Args&... args) {//任意個のchat・string・int等を引数に取り文字列として繋げたstringを返す(整数はasciiでcharに変換)
	string res;
	(res += ... += args);
	return res;
}

//その他
ll max(ll a, int b) { return max(a, ll(b)); }
ll max(int a, ll b) { return max(ll(a), b); }
ll min(ll a, int b) { return min(a, ll(b)); }
ll min(int a, ll b) { return min(ll(a), b); }
bool yn(bool jg) {
	if (jg) {
		cout << "Yes" << endl; return true;
	}
	else {
		cout << "No" << endl; return false;
	}
}
bool ynab(bool jg) {
	if (jg) {
		cout << "Alice" << endl; return true;
	}
	else {
		cout << "Bob" << endl; return false;
	}
}
template<typename T>
void cp(map < T, ll >& a, T b, ll c) {//countplus
	if (a.find(b) == a.end()) a[b] = c;
	else a[b] += c;
	if (a[b] == 0) a.erase(b);
}
template<typename T>
bool cross(T a, T b, T c, T d) {//区間(a,b)と区間(c,d)が重なるか判定
	return !(b <= c || d <= a);
}
template<typename T>
bool crosse(T a, T b, T c, T d) {//区間[a,b]と区間[c,d]が重なるか判定
	return !(b < c || d < a);
}
template<typename T>
vec<vll> absort(vec<T> a, vec<T> b) {//l,rのl昇順r逆順sort
	vec<vec<T>> ab(a.sz);
	rp(i, a.sz) ab[i] = { a[i],-b[i],i };
	sort(all(ab));
	rp(i, a.sz) ab[i][1] = -ab[i][1];
	return ab;
}
template<typename T>
bool rabc(T a, T b, T c) {//円環状でa,b,cがこの順か判定 同じ値はfalse
	return ((a - b) / abs(a - b)) * ((b - c) / abs(b - c)) * ((c - a) / abs(c - a)) > 0;
}
bool labc(ll a, ll b, ll c) {//a<=b<cか判定
	return a <= b && b < c;
}
#define inr labc
bool labc2(ll a, ll b, ll c) {//a<=b<=cか判定
	return a <= b && b <= c;
}
#define inr2 labc2

vector<vector<char>> rot(vs a) {//時計回り90度回転
	vector<vector<char>> r(a[0].sz, vector<char>(a.sz));
	rp(i, a.sz)rp(j, a[0].sz) r[j][a[0].sz - 1 - i] = a[i][j];
	return r;
}
template<typename T>
vector<vector<T>> rot(vector<vector<T>> a) {
	vector<vector<T>> r(a[0].sz, vector<T>(a.sz));
	rp(i, a.sz)rp(j, a[0].sz) r[j][a.sz - 1 - i] = a[i][j];
	return r;
}


//実行時間調整付きassert
std::chrono::steady_clock::time_point START_TIME;
void start() {
	START_TIME = std::chrono::steady_clock::now();
}

void assertt(bool b, int t) {
	if (b) return;
	if (START_TIME.time_since_epoch().count() == 0) {
		start();
	}
	auto target_time = START_TIME + std::chrono::milliseconds(t - 100);
	while (std::chrono::steady_clock::now() < target_time);
	assert(0);
}
bool testt(int t) {//tミリ秒経っていないならtrue (while(testt(1950))のように使う)
	assert(START_TIME.time_since_epoch().count() != 0);
	auto target_time = START_TIME + std::chrono::milliseconds(t - 100);
	return (std::chrono::steady_clock::now() < target_time);
}
//lazy_segtree<S, op, e, F, mapping, composition, id> seg(n);
//S:中身型 op:左右合成 e:初期値
//F:遅延型 mapping:遅延適用後の要素 composition:遅延合成( 関数f(g()) ) id:遅延初期値(何もさせない)


using S = pll;
using F = ll;
S op(S a, S b) {
	return {a.fi+b.fi,a.se+b.se};
}

S e() {
	return { 0,0 };
}
using segt = segtree<S, op, e>;
S mapping(F x, S a) {//xをaに作用(x初期値に注意) S[l]~S[r]のopであるSにF適用するとF(S[l])~F(S[r])のopになる必要がある 、つまりL~Rの代表にFかけてもL~Rの代表であり続けられるようS作る必要がある
	return { a.fi + a.se * x,a.se };
}
F composition(F f, F g) {//fの方が後(f初期値に注意) Fを何個合成しても1つのFで表せる
	return f + g;
}
F id() { return 0; }
using lsegt = lazy_segtree<S, op, e, F, mapping, composition, id>;

struct HLD {
	int n;
	vi pa, depth, sb, hch;//sb:部分木の頂点数 hch:heavy child
	vvi g, ch;
	vi v;//v[i]=heavy child優先でDFSしてi番目に訪れる頂点 heavy pathはv上で一列に並んでいる
	vi vid;//id[i]=heavy child優先でDFSして頂点iを何番目に訪れるか(vの逆関数)
	vi head;//head[i]=頂点iを通るheavy pathの一番上の頂点
	static pair<S, S> op2(pair<S, S> a, pair<S, S> b) { return { op(a.fi,b.fi),op(b.se,a.se) }; }
	static pair<S, S> e2() { return { e(),e() }; }
	//segtree<pair<S,S>, op2, e2> seg;
	static pair<S, S> mapping2(F x, pair<S, S> a) { return { mapping(x,a.fi),mapping(x,a.se) }; }
	lazy_segtree<pair<S, S>, op2, e2, F, mapping2, composition, id> seg;

	int edge_cnt; bool is_built;
	HLD(int n) : n(n), g(n), pa(n, -1), ch(n), depth(n, 0), sb(n, 1), hch(n, -1), v(n), vid(n), head(n), seg(n), edge_cnt(0), is_built(0) {}
	void add_edge(int u, int v) {
		edge_cnt++;
		assert(0 <= u && u < n && 0 <= v && v < n);
		g[u].push_back(v);
		g[v].push_back(u);
	}
	void build() {
		assert(edge_cnt == n - 1 && is_built == 0);
		is_built = 1;
		auto dfs0 = [&](auto& self, int s)->void {
			int h = -1;//現在のheavy child候補を根とする部分木の頂点数
			for (auto i : g[s]) if (i != pa[s]) {//sの子i全てに対して
				pa[i] = s;
				ch[s].pb(i);
				depth[i] = depth[s] + 1;
				self(self, i);
				sb[s] += sb[i];
				if (chmax(h, sb[i])) hch[s] = i;
			}
			};
		int cnt = 0;//今までに訪れた頂点数
		auto dfs1 = [&](auto& self, int s)->void {
			v[cnt] = s;
			vid[s] = cnt;
			cnt++;
			if (s != 0 && hch[pa[s]] == s) head[s] = head[pa[s]];
			else head[s] = s;
			if (hch[s] != -1)self(self, hch[s]);
			for (auto i : g[s]) if (i != pa[s] && i != hch[s]) {
				self(self, i);
			}
			};
		dfs0(dfs0, 0);
		dfs1(dfs1, 0);
	}
	int pn(int a, int x) {//aのx個上の先祖 存在しないなら-1 O(logN)
		assert(0 <= a && a < n && 0 <= x && is_built);
		int td = depth[a] - x;
		if (td < 0) return -1;
		while (a != -1) {
			if (depth[head[a]] <= td) return v[vid[head[a]] + td - depth[head[a]]];
			a = pa[head[a]];
		}
		return -1;
	};
	int lca(int a, int b) {//a,bのlca O(logN)
		assert(0 <= a && a < n && 0 <= b && b < n && is_built);
		while (head[a] != head[b]) {
			if (vid[a] > vid[b]) swap(a, b);
			b = pa[head[b]];
		}
		if (vid[a] > vid[b]) return b;
		return a;
	}
	int dist1(int a, int b) {//頂点aからbの間の辺の数
		assert(is_built);
		return depth[a] + depth[b] - 2 * depth[lca(a, b)];
	}
	int jump(int a, int b, int k) {//パス{a,...,b}におけるk番目の頂点(k==0のときa) 存在しなければ-1
		assert(is_built);
		int l = lca(a, b);
		int dist = depth[a] + depth[b] - 2 * depth[l];
		if (dist < k) return -1;
		if (depth[a] - depth[l] >= k) {
			return pn(a, k);
		}
		else return pn(b, dist - k);
	}
	void sset(int i, S x) {//頂点iにxをset
		assert(is_built);
		seg.set(vid[i], { x,x });
	}
	S sget(int i) {//頂点iをget
		return seg.get(vid[i]).fi;
	}
	S sprod(int a, int b) {//パス{a,...,b}に対してこの順での総opを計算 (半開区間ではない)
		S ans_up = e();
		S ans_down = e();
		while (head[a] != head[b]) {
			if (depth[head[a]] > depth[head[b]]) {
				ans_up = op(ans_up, seg.prod(vid[head[a]], vid[a] + 1).se);
				a = pa[head[a]];
			}
			else {
				ans_down = op(seg.prod(vid[head[b]], vid[b] + 1).fi, ans_down);
				b = pa[head[b]];
			}
		}
		if (depth[a] > depth[b]) {
			ans_up = op(ans_up, seg.prod(vid[b], vid[a] + 1).se);
		}
		else {
			ans_down = op(seg.prod(vid[a], vid[b] + 1).fi, ans_down);
		}
		return op(ans_up, ans_down);
	}
	void sapply(int a, int b, F f) {//パス{a,...,b}に対してfをapply (半開区間ではない)
		while (head[a] != head[b]) {
			if (depth[head[a]] > depth[head[b]]) {
				seg.apply(vid[head[a]], vid[a] + 1, f);
				a = pa[head[a]];
			}
			else {
				seg.apply(vid[head[b]], vid[b] + 1, f);
				b = pa[head[b]];
			}
		}
		if (depth[a] > depth[b]) {
			seg.apply(vid[b], vid[a] + 1, f);
		}
		else {
			seg.apply(vid[a], vid[b] + 1, f);
		}
		return;
	}
};
int main() {

}