#include <stdio.h>
#include <sstream>
#include <string.h>
#include <vector>
#include <map>
#include <algorithm>
#include <utility>
#include <set>
#include <cctype>
#include <queue>
#include <stack>
#include <cstdio>
#include <cstdlib>
#include <cmath>
#include <deque>
#include <limits>
#include <iomanip>
#include <ctype.h>
#include <unordered_map>
#include <random>
#include <numeric>
#include <iostream>
#include <array>
#include <atcoder/all>
#include <functional>

#define _USE_MATH_DEFINES
#include <iostream>
#include <fstream>
#include <math.h>
#include <bitset>
#pragma intrinsic(_umul128)
using namespace std;
using namespace atcoder;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;
typedef pair<ll, double> pld;
typedef pair<double, double> pdd;
typedef pair<double, ll> pdl;
typedef pair<int, char> pic;
typedef vector<ll> vl;
typedef vector<double> vd;
typedef vector<ull> vul;
typedef vector<pll> vpll;
typedef vector<int> vi;
typedef vector<string> table;
typedef priority_queue<ll, vector<ll>, greater<ll>> llgreaterq;
typedef priority_queue<pll, vector<pll>, greater<pll>> pllgreaterq;
typedef priority_queue<pair<ll, pll>, vector<pair<ll, pll>>, greater<pair<ll, pll>>> plpllgreaterq;
typedef priority_queue<vi, vector<vi>, greater<vi>> vigreaterq;
typedef priority_queue<vl, vector<vl>, greater<vl >> vlgreaterq;
typedef vector<vl> mat;
typedef vector<mat> thd;
typedef modint1000000007 mint7;
typedef modint998244353 mint9;
typedef modint mint;
template <class o, class p>
using pairq = priority_queue<pair<o, p>, vector<pair<o, p>>, greater<pair<o, p>>>;
template <class o, class p, class q>
using tuple3q = priority_queue<tuple<o, p, q>, vector<tuple<o, p, q>>, greater<tuple<o, p, q>>>;
template <class o, class p, class q, class r>
using tuple4q = priority_queue<tuple<o, p, q, r>, vector<tuple<o, p, q, r>>, greater<tuple<o, p, q, r>>>;
template <class o, class p, class q, class r, class s>
using tuple5q = priority_queue<tuple<o, p, q, r, s>, vector<tuple<o, p, q, r, s>>, greater<tuple<o, p, q, r, s>>>;

vl dx = { 1,0,-1,0 };
vl dy = { 0,1,0,-1 };
int dxe[] = { 1,1,0,-1,-1,-1,0,1 };
int dye[] = { 0,1,1,1,0,-1,-1,-1 };
#define bit(x,v) ((ll)x << v)
#define rep(x,n) for(ll x = 0;x < n;x++)
#define rep2(x,f,v) for(ll x=f;x<v;x++)
#define repe(v,x) for(auto v : x)
// 許容する誤差ε
#define EPS (1e-10)
// 2つのスカラーが等しいかどうか
#define EQ(a,b) (std::abs(a-b) < EPS)
// 2つのベクトルが等しいかどうか
#define EQV(a,b) ( EQ((a).real(), (b).real()) && EQ((a).imag(), (b).imag()) )
#define all(a) a.begin(),a.end()
#define all0(a) memset(a,0,sizeof(a))
#define allm1(a) memset(a,-1,sizeof(a))
#define set_float() cout << fixed << setprecision(12);
#define coutl(s) cout <<s <<endl
#define pln(s) cout<<s<<"\n"
#define ple pln(-1)
#define plm(s) cout<<(s).val()<<"\n"
#define plm17(s) cout<<modint1000000007(s).val()<<"\n"
#define plm9(s) cout<<modint998244353(s).val()<<"\n"
#define put_float(v) 	set_float() \
						pln(v)
#define vinsert(v,p,x) v.insert(v.begin() + p,x)
#define vsort(v) sort(all(v));
#define vdesc(v) vsort(v); \
					reverse(all(v))
#define gete(u,v) ll u,v; cin>>u>>v; u--;v--;
#define getpair(a,b) ll a,b;cin>>a>>b;
#define dup(v) v.erase(unique(all(v)),v.end())
#define cub(a) (a)*(a)*(a)
#define ion(i,j) (i & (1LL << j))
#define Len size()
#define psp(a,b) push_back(make_pair(a,b))
#define psp2(a,b) push(make_pair(a,b))
#define cini(a) a; cin >> a
#define infa(a,b) (a + b) % INF
#define infm(a,b) (a * b) % INF
#define infd(a,b) (a * INFinv(b)) % INF
#define infs(a,b) (a + INF - inff(b)) % INF
#define inf(a) (a) %= INF
#define inff(a) ((a + INF) % INF)
#define No cout << "No" << endl
#define Yes cout << "Yes" << endl
#define NO cout << "NO" << endl
#define YES cout << "YES" << endl
#define errm1 pln(-1);return;
#define smal -(ll)1000000009*1000000009
#define big (ll)1000000009*1000000009
#define frontpop(q) q.front();q.pop()
#define toppop(q) q.top();q.pop()
#define arr(a,s) a[s]; all0(a);
#define nxt(cu) (cu+1) % 2
#define chkover(x,y,h,w) (x<0||y<0||x>=h||y>=w)
#define psb(v) ll value;cin>>value;v.push_back(value);
#define lower_b(v,p) lower_bound(all(v), p)
#define upper_b(v,p) upper_bound(all(v), p)
#define allpln(v) for(auto &e:v)pln(e)
#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define msize 216;
#define revarr(p,l,r) reverse(p.begin()+l,p.begin()+r+1)
#define reverse_all(p) reverse(all(p))
#define cill(x) ll x;cin>>x
#define cilll(x,y) ll x,y;cin>>x>>y
#define bitn(x,k)(((x)>>(k))&1)
#define iotan(a,n) iota(all(a),n)
#define cline(a,k) vl a(k); rep(i,k){cin>>a[i];}
#define clineu(a,k) vul a(k); rep(i,k){cin>>a[i];}
#define clines(a,k) vector	<string> a(k); rep(i,k){cin>>a[i];}
#define cindec(a) ll a; cin>> a; a--;
template <typename T, typename U>
T SUM(const vector<U>& A) {
	T sum = 0;
	for (auto&& a : A) sum += a;
	return sum;
}

ll ceil(ll a, ll b) { return a > 0 ? (a - 1) / b + 1 : a / b; }

ll n, m;

bool chmin(ll& a, ll b) {
	if (a > b) {
		a = b; return 1;
	}
	return 0;
}
bool chmind(double& a, double b) {
	if (a > b) {
		a = b; return 1;
	}
	return 0;
}
template<class T> inline bool chmax(T& a, T b) { if (a < b) { a = b; return 1; } return 0; }
ll INF = 1000000007;
const int MAX = 3000010;
void cout2(ll val) {
	if (val >= big) {
		pln(-1);
	}
	else {
		pln(val);
	}
}
void cout3(ll val) {
	if (val >= INF) {
		pln(-1);
	}
	else {
		pln(val);
	}
}
template <typename T>
vector<T> merge_arr(vector<T>& a, vector<T>& b) {
	vector<T> c(a.size() + b.size());
	std::merge(all(a), all(b), c.begin());
	return c;
}
string padleft(string x, ll dig, char c) {
	ll si = x.size();
	for (ll i = 0; i < dig - si; i++)
	{
		x = c + x;
	}
	return x;
}
long long fac[MAX], finv[MAX], inv[MAX], called;
void COMinit() {
	fac[0] = fac[1] = 1;
	finv[0] = finv[1] = 1;
	inv[1] = 1;
	for (int i = 2; i < MAX; i++) {
		fac[i] = fac[i - 1] * i % INF;
		inv[i] = INF - inv[INF % i] * (INF / i) % INF;
		finv[i] = finv[i - 1] * inv[i] % INF;
	}
}
void COMinit998244353() {
	INF = 998244353;
	COMinit();
	called = 1;
}
void COMinit1000000007() {
	INF = 1000000007;
	COMinit();
	called = 1;
}

ll gfac(ll x) {
	if (!called) {
		COMinit();
		called = 1;
	}
	return fac[x];
}
// 二項係数計算
long long COM(int n, int k) {
	if (!called) {
		COMinit();
		called = 1;
	}
	if (n < k) return 0;
	if (n < 0 || k < 0) return 0;
	return fac[n] * (finv[k] * finv[n - k] % INF) % INF;
}

modint998244353 COM2(ll n, ll k) {
	modint998244353 res = 1;
	rep(i, k) {
		res *= (n - i);
		res /= (i + 1);
	}
	return res;
}
ll getpow(ll b, ll x, ll md) {
	ll t = b % md;

	ll res = 1;
	while (x > 0)
	{
		if (x & 1) {
			res *= t;
			res %= md;
		}
		x >>= 1;
		t *= t;
		t %= md;
	}
	return res % md;
}
ull getpowul(ull b, ull x, ull md) {
	ull t = b % md;

	ull res = 1;
	while (x > 0)
	{
		if (x & 1) {
			res *= t;
			res %= md;
		}
		x >>= 1;
		t *= t;
		t %= md;
	}
	return res % md;
}
ll getpow(ll b, ll x) {
	return getpow(b, x, INF);
}
/// 素数を法とする場合
ll modinv(ll x) {
	return getpow(x, INF - 2);
}

ll extgcd(ll a, ll b, ll& x, ll& y) {
	ll d = a;
	if (b != 0) {
		d = extgcd(b, a % b, y, x);
		y -= (a / b) * x;
	}
	else {
		x = 1; y = 0;
	}
	return d;
}

/// <summary>
/// 素数を法としない場合
/// </summary>
/// <param name="a"></param>
/// <param name="m"></param>
/// <returns></returns>
ll modinv(ll a, ll m) {
	ll x, y;
	extgcd(a, m, x, y);
	return (m + x % m) % m;
}

ll gcd(ll a, ll b) {
	if (b == 0) return a;
	return gcd(b, a % b);
}
class m_random {
	std::mt19937 mt;
	std::uniform_int_distribution<> rand100;
public:
	m_random(ll mi, ll ma) {
		init_random(mi, ma);
	}
	void init_random(ll mi, ll ma) {
		std::random_device rnd;     // 非決定的な乱数生成器を生成
		mt = std::mt19937(rnd());     //  メルセンヌ・ツイスタの32ビット版、引数は初期シード値
		rand100 = std::uniform_int_distribution<>(mi, ma);
	}
	ll get() {
		return rand100(mt);
	}

};

class m_sampling {
	std::mt19937 mt;
	std::normal_distribution<double> rand;
public:
	m_sampling(double sigma) {
		init_sampling(sigma);
	}
	void init_sampling(double sigma) {
		std::random_device rnd;
		mt = std::mt19937(rnd());
		rand = std::normal_distribution<double>(0.0, sigma);
	}
	double get() {
		return rand(mt);
	}
};

typedef vector<modint998244353> vml;
typedef vector<vml> matm;
typedef vector<modint1000000007> vml2;
typedef vector<vml2> matm2;
typedef vector<modint> vml3;
typedef vector<vml3> matm3;
#define cmat(n,s,ss) mat n(s,vl(ss))
#define cmatm(n,s,ss) matm n(s,vml(ss))
#define cmatm2(n,s,ss) matm2 n(s,vml2(ss))
#define cmatm3(n,s,ss) matm3 n(s,vml3(ss))

// Union find
vl pr;
vl lank;
vl udpt;
void uini(int _n) {
	_n++; // 一個拡張しておく
	pr = vl(_n + 1);
	lank = vl(_n + 1);
	udpt = vl(_n + 1, 0);
	for (ll i = 0; i <= _n; i++)
	{
		pr[i] = i;
		lank[i] = 1;
	}
}

int parent(int x) {
	if (x == pr[x]) return x;
	auto paren = parent(pr[x]);
	udpt[x] = udpt[paren] + 1;
	return pr[x] = paren;
}

int same(int x, int y) {
	return parent(x) == parent(y);
}

bool unit(int x, int y) {
	int px = parent(x);
	int py = parent(y);

	if (px == py) return false;
	if (lank[py] <= lank[px]) {
		pr[py] = px;
		lank[px] += lank[py];
	}
	else {
		pr[px] = py;
		lank[py] += lank[px];
	}
	return true;
}

ll unisize(ll i) {
	return lank[parent(i)];
}
bool unitm(int x, int y) {
	int px = parent(x);
	int py = parent(y);

	if (px == py) return false;
	if (lank[py] < lank[px]) {
		pr[py] = px;
		lank[px] += lank[py];
	}
	else {
		pr[px] = py;
		lank[py] += lank[px];
	}
	return true;
}
/// <summary>
/// 数字の小さい方を親にするように処理
/// </summary>
/// <param name="x"></param>
/// <param name="y"></param>
/// <returns></returns>
bool unitlow(int x, int y) {
	int px = parent(x);
	int py = parent(y);

	if (px == py) return false;

	if (py < px) {
		pr[py] = px;
		lank[px] += lank[py];
	}
	else {
		pr[px] = py;
		lank[py] += lank[px];
	}
	return true;
}

ll clamp(ll t, ll l, ll r) {
	return max(l, min(r, t));
}

int H;
int left(int i) {
	return i * 2 + 1;
}
int right(int i) {
	return i * 2 + 2;
}
class edge {
public:
	int from, to, i;
	ll val;
	ll cap, rev, icap;
	edge() {}
	edge(ll to) : to(to) {}
	edge(ll to, ll i) : to(to), i(i) {}
	edge(ll from, ll to, ll val) : from(from), to(to), val(val) {}
	void flowEdge(ll _to, ll _cap, ll _rev) {
		to = _to;
		cap = _cap;
		icap = _cap;
		rev = _rev;
	}
};
typedef vector<vector<edge>> vve;

class LCA {
private:
	vector<vector<edge>> v;
	vector<vector<int>> parent;
	vector<int> depth;
	ll root;
	void dfs(int n, int m, int d) {
		parent[0][n] = m;
		depth[n] = d;
		for (auto x : v[n]) {
			if (x.to != m) dfs(x.to, n, d + 1);
		}
	}
public:
	LCA() {}
	LCA(ll N, ll root, vector<vector<edge>>& tree) {
		v = tree;
		this->root = root;
		parent = vector<vector<int>>(21, vector<int>(N + 1, 0));
		depth = vector<int>(N + 1, 0);
		dfs(root, -1, 0);
		for (int j = 0; j + 1 < 20; j++) {
			for (int i = 1; i <= N; i++) {
				if (parent[j][i] < 0) parent[j + 1][i] = -1;
				else parent[j + 1][i] = parent[j][parent[j][i]];
			}
		}
	}
	int lca(int n, int m) {
		if (depth[n] > depth[m]) swap(n, m);
		if (n == root)
			return root;
		for (int j = 0; j < 20; j++) {
			if ((depth[m] - depth[n]) >> j & 1) m = parent[j][m];
		}
		if (n == m) return n;
		for (int j = 19; j >= 0; j--) {
			if (parent[j][n] != parent[j][m]) {
				n = parent[j][n];
				m = parent[j][m];
			}
		}
		return parent[0][n];
	}
	int dep(int n) { return depth[n]; }
};
ll k;
int _rank[1010];
int temp[1010];
bool compare_sa(int i, int j) {
	if (_rank[i] != _rank[j]) return _rank[i] < _rank[j];
	else {
		int ri = i + k <= n ? _rank[i + k] : -1;
		int rj = j + k <= n ? _rank[j + k] : -1;
		return ri < rj;
	}
}
void construct_sa(string S, int* sa) {
	n = S.length();

	for (ll i = 0; i <= n; i++)
	{
		sa[i] = i;
		_rank[i] = i < n ? S[i] : -1;
	}

	for (k = 1; k <= n; k *= 2)
	{
		sort(sa, sa + n + 1, compare_sa);

		// saはソート後の接尾辞の並びになっている、rankは元の位置のindexを保持したまま、更新されている。
		// ピンとこなかった部分
		temp[sa[0]] = 0;
		for (ll i = 1; i <= n; i++)
		{
			temp[sa[i]] = temp[sa[i - 1]] + (compare_sa(sa[i - 1], sa[i]) ? 1 : 0);
		}
		for (ll i = 0; i <= n; i++)
		{
			_rank[i] = temp[i];
		}
	}
}
bool contain(string S, int* sa, string T) {
	int a = 0, b = S.length();
	// sa は 接尾辞が辞書順に並んでいる、入っているのはその位置のインデックス
	while (b - a > 1) {
		int c = (a + b) / 2;
		if (S.compare(sa[c], T.length(), T) < 0) a = c;
		else b = c;
	}
	return S.compare(sa[b], T.length(), T) == 0;
}

#define bit(x,v) ((ll)x << v)

class BIT {
	static const int MAX_N = 500010;
public:
	vl bit;
	ll n;
	BIT() { bit = vl(MAX_N + 1, 0); }
	BIT(ll _n) {
		bit = vl(_n * 2 + 10, 0);
		n = _n;
	}
	ll sum(int i) {
		ll s = 0;
		while (i > 0)
		{
			s += bit[i];
			i -= i & -i;
		}
		return s;
	}

	void add(int i, int x) {
		while (i <= n)
		{
			bit[i] += x;
			i += i & -i;
		}
	}
};
struct UnionFind {
	vector<int> A;
	UnionFind(int n) : A(n, -1) {}
	int find(int x) {
		if (A[x] < 0) return x;
		return A[x] = find(A[x]);
	}
	void unite(int x, int y) {
		x = find(x), y = find(y);
		if (x == y) return;
		if (A[x] > A[y]) swap(x, y);
		A[x] += A[y];
		A[y] = x;
	}
	int ngroups() {
		int ans = 0;
		for (auto a : A) if (a < 0) ans++;
		return ans;
	}
};
vector<ll> getp(ll n) {
	vector<ll> res;
	if (n % 2 == 0) {
		res.push_back(2);
		while (n % 2 == 0)n /= 2;
	}

	for (ll i = 3; i * i <= n; i += 2)
	{
		if (n % i == 0) {
			res.push_back(i);
			while (n % i == 0)n /= i;
		}
	}
	if (n != 1) res.push_back(n);
	return res;
}
vector<ll> getpp(ll n) {
	vector<ll> res;
	if (n % 2 == 0) {
		res.push_back(2);
		while (n % 2 == 0)n /= 2;
	}

	for (ll i = 3; i * i * i <= n; i += 2)
	{
		if (n % i == 0) {
			res.push_back(i);
			while (n % i == 0)n /= i;
		}
	}
	if (n != 1) res.push_back(n);
	return res;
}
vector<ll> getp2(ll n) {
	vector<ll> res;
	if (n % 2 == 0) {
		while (n % 2 == 0) { n /= 2; res.push_back(2); }
	}

	for (ll i = 3; i * i <= n; i += 2)
	{
		if (n % i == 0) {
			while (n % i == 0) { n /= i; res.push_back(i); }
		}
	}
	if (n != 1) res.push_back(n);
	return res;
}
vector<pll> getp3(ll n) {
	vector<pll> res;
	int si = 0;
	if (n % 2 == 0) {
		res.push_back(make_pair(2, 0));
		while (n % 2 == 0) { n /= 2; res[si].second++; }
		si++;
	}

	for (ll i = 3; i * i <= n; i += 2)
	{
		if (n % i == 0) {
			res.push_back(make_pair(i, 0));
			while (n % i == 0) { n /= i; res[si].second++; }
			si++;
		}
	}
	if (n != 1) { res.push_back(make_pair(n, 1)); }
	return res;
}

vector<ll> getDivisors(ll n) {
	vector<ll> res;
	res.push_back(1);
	if (1 < n)
		res.push_back(n);
	for (ll i = 2; i * i <= n; i++)
	{
		if (n % i == 0) {
			res.push_back(i);
			if (n / i != i)
				res.push_back(n / i);
		}
	}
	vsort(res);
	return res;
}

struct ve {
public:
	vector<ve> child;
	int _t = INF;
	ve(int t) :_t(t) {}
	ve(ve _left, ve _right) {
		_t = _left._t + _right._t;
		child.push_back(_left);
		child.push_back(_right);
	}
	bool operator<(const ve& t) const {
		return _t > t._t;
	}
};

vector<bool> elas(ll n) {
	n++;
	vector<bool> r(n, 1);
	r[0] = 0;
	r[1] = 0;
	ll tw = 4;
	while (tw < n) {
		r[tw] = false;
		tw += 2;
	}
	ll th = 6;
	while (th < n) {
		r[th] = false;
		th += 3;
	}
	ll fv = 10;
	while (fv < n) {
		r[fv] = false;
		fv += 5;
	}

	for (ll i = 6; i * i < n; i += 6)
	{
		ll bf = i - 1;
		if (r[bf]) {
			ll ti = bf * 2;
			while (ti < n)
			{
				r[ti] = false;
				ti += bf;
			}
		}
		ll nx = i + 1;
		if (r[nx]) {
			ll ti = nx * 2;
			while (ti < n)
			{
				r[ti] = false;
				ti += nx;
			}
		}
	}
	return r;
}

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

class SegTree {
public:
	const static int MAX_N = 1000100;
	const static int DAT_SIZE = (1 << 20) - 1;
	int N, Q;
	int A[MAX_N];
	ll MAX = big;

	ll data[DAT_SIZE], datb[DAT_SIZE];
	void init(int _n) {
		N = 1;
		while (N < _n) N <<= 1;
		memset(data, 0, sizeof(data));
		memset(datb, 0, sizeof(datb));
	}
	void init(int _n, ll iv) {
		N = 1;
		while (N < _n) N <<= 1;
		rep(i, DAT_SIZE) {
			data[i] = iv;
			datb[i] = iv;
		}
	}
	void initRMQ(int _n) {
		N = 1;
		while (N < _n) N *= 2;
		// 全ての値をbigに
		rep(i, 2 * N - 1)
			data[i] = MAX;
	}
	void updateRMQ(int k, ll a) {
		k += N - 1;
		data[k] = a;
		while (k > 0) {
			k = (k - 1) / 2;
			data[k] = min(data[k * 2 + 1], data[k * 2 + 2]);
		}
	}
	ll RMQ(int a, int b) {
		return queryRMQ(a, b + 1, 0, 0, N);
	}
	ll queryRMQ(int a, int b, int k, int l, int r) {
		if (r <= a || b <= l)
			return MAX;

		// [a,b)が[l,r)を完全に含んでいれば
		if (a <= l && r <= b)
			return data[k];

		// そうでなければ２つの子の最小値
		// n=16
		// 0,16→0,8 8,16
		// 0,4 4,8 8,12 12,16
		ll vl = queryRMQ(a, b, k * 2 + 1, l, (l + r) / 2);
		ll vr = queryRMQ(a, b, k * 2 + 2, (l + r) / 2, r);
		return min(vl, vr);
	}

	void add(int a, int b, int x) {
		add(a, b + 1, x, 0, 0, N);
	}
	void add(int a, int b, int x, int k, int l, int r) {
		if (a <= l && r <= b) {
			data[k] += x;
		}
		else if (l < b && a < r) {
			datb[k] += (min(b, r) - max(a, l)) * x;
			add(a, b, x, k * 2 + 1, l, (l + r) / 2);
			add(a, b, x, k * 2 + 2, (l + r) / 2, r);
		}
	}

	void change(int a, int b, int x) {
		change(a, b + 1, x, 0, 0, N);
	}
	void change(int a, int b, int x, int k, int l, int r) {
		if (a <= l && r <= b) {
			data[k] = x;
		}
		else if (l < b && a < r) {
			datb[k] = x;
			change(a, b, x, k * 2 + 1, l, (l + r) / 2);
			change(a, b, x, k * 2 + 2, (l + r) / 2, r);
		}
	}

	ll sum(int a, int b) {
		return sum(a, b + 1, 0, 0, N);
	}
	ll sum(int a, int b, int k, int l, int r) {
		if (b <= l || r <= a) {
			return 0;
		}
		if (a <= l && r <= b) {
			return data[k] * (r - l) + datb[k];
		}

		ll res = (min(b, r) - max(a, l)) * data[k];
		res += sum(a, b, k * 2 + 1, l, (l + r) / 2);
		res += sum(a, b, k * 2 + 2, (l + r) / 2, r);
		return res;
	}
};

class LazySegTree {
private:
	int N;
	vl node, lazy;
public:
	void init(int _n) {
		N = 1;
		while (N < _n) N <<= 1;
		node.resize(2 * N, 0);
		lazy.resize(2 * N, 0);
	}

	// k 番目のノードについて遅延評価を行う
	void eval(int k, int l, int r) {
		// 遅延配列が空でない場合、自ノード及び子ノードへの
		// 値の伝播が起こる
		if (lazy[k] != 0) {
			node[k] += lazy[k];

			// 最下段かどうかのチェックをしよう
			// 子ノードは親ノードの 1/2 の範囲であるため、
			// 伝播させるときは半分にする
			if (r - l > 1) {
				lazy[2 * k + 1] += lazy[k] / 2;
				lazy[2 * k + 2] += lazy[k] / 2;
			}

			// 伝播が終わったので、自ノードの遅延配列を空にする
			lazy[k] = 0;
		}
	}
	void add(int a, int b, ll x) {
		addbody(a, b + 1, x);
	}
	void addbody(int a, int b, ll x, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = N;

		// k 番目のノードに対して遅延評価を行う
		eval(k, l, r);

		// 範囲外なら何もしない
		if (b <= l || r <= a) return;

		// 完全に被覆しているならば、遅延配列に値を入れた後に評価
		if (a <= l && r <= b) {
			lazy[k] += (r - l) * x;
			eval(k, l, r);
		}

		// そうでないならば、子ノードの値を再帰的に計算して、
		// 計算済みの値をもらってくる
		else {
			addbody(a, b, x, 2 * k + 1, l, (l + r) / 2);
			addbody(a, b, x, 2 * k + 2, (l + r) / 2, r);
			node[k] = node[2 * k + 1] + node[2 * k + 2];
		}
	}

	ll getsum(int a, int b, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = N;
		if (b <= l || r <= a) return 0;

		// 関数が呼び出されたら評価！
		eval(k, l, r);
		if (a <= l && r <= b) return node[k];
		ll vl = getsum(a, b, 2 * k + 1, l, (l + r) / 2);
		ll vr = getsum(a, b, 2 * k + 2, (l + r) / 2, r);
		return vl + vr;
	}

	ll getMax(int a, int b) {
		// 半開区間に変換
		return getMaxBdy(a, b + 1);
	}

	ll getMaxBdy(int a, int b, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = N;
		if (b <= l || r <= a) return -big;

		// 関数が呼び出されたら評価！
		eval(k, l, r);
		if (a <= l && r <= b) return node[k];
		ll vl = getMaxBdy(a, b, 2 * k + 1, l, (l + r) / 2);
		ll vr = getMaxBdy(a, b, 2 * k + 2, (l + r) / 2, r);
		return max(vl, vr);
	}
};

class LazySegTreeRMQ {
private:
	int N;
	vl node, lazy;
public:
	void init(int _n) {
		N = 1;
		while (N < _n) N <<= 1;
		node.resize(2 * N, 0);
		lazy.resize(2 * N, 0);
	}

	// k 番目のノードについて遅延評価を行う
	void eval(int k, int l, int r) {
		if (lazy[k] != 0) {
			node[k] = lazy[k];

			if (r - l > 1) {
				lazy[2 * k + 1] = lazy[k];
				lazy[2 * k + 2] = lazy[k];
			}

			lazy[k] = 0;
		}
	}
	void evalAdd(int k, int l, int r) {
		if (lazy[k] != 0) {
			node[k] += lazy[k];

			if (r - l > 1) {
				lazy[2 * k + 1] += lazy[k];
				lazy[2 * k + 2] += lazy[k];
			}

			lazy[k] = 0;
		}
	}
	void add(int a, int b, ll x) {
		addbody(a, b + 1, x);
	}
	void addbody(int a, int b, ll x, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = N;

		// k 番目のノードに対して遅延評価を行う
		evalAdd(k, l, r);

		// 範囲外なら何もしない
		if (b <= l || r <= a) return;

		// 完全に被覆しているならば、遅延配列に値を入れた後に評価
		if (a <= l && r <= b) {
			lazy[k] += x;
			evalAdd(k, l, r);
		}

		// そうでないならば、子ノードの値を再帰的に計算して、
		// 計算済みの値をもらってくる
		else {
			addbody(a, b, x, 2 * k + 1, l, (l + r) / 2);
			addbody(a, b, x, 2 * k + 2, (l + r) / 2, r);
			node[k] = max(node[2 * k + 1], node[2 * k + 2]);
		}
	}

	void update(int a, int b, ll v) {
		updateBdy(a, b + 1, v);
	}

	void updateBdy(int a, int b, ll x, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = N;

		// k 番目のノードに対して遅延評価を行う
		eval(k, l, r);

		// 範囲外なら何もしない
		if (b <= l || r <= a) return;

		// 完全に被覆しているならば、遅延配列に値を入れた後に評価
		if (a <= l && r <= b) {
			if (x > node[k]) {
				lazy[k] = x;
				eval(k, l, r);
			}
		}

		// そうでないならば、子ノードの値を再帰的に計算して、
		// 計算済みの値をもらってくる
		else {
			updateBdy(a, b, x, 2 * k + 1, l, (l + r) / 2);
			updateBdy(a, b, x, 2 * k + 2, (l + r) / 2, r);
			node[k] = max(node[2 * k + 1], node[2 * k + 2]);
		}
	}

	ll getMaxAdd(int a, int b) {
		// 半開区間に変換
		return getMaxAddBdy(a, b + 1);
	}
	ll getMaxAddBdy(int a, int b, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = N;
		if (b <= l || r <= a) return -big;

		// 関数が呼び出されたら評価！
		evalAdd(k, l, r);
		if (a <= l && r <= b) return node[k];
		ll vl = getMaxAddBdy(a, b, 2 * k + 1, l, (l + r) / 2);
		ll vr = getMaxAddBdy(a, b, 2 * k + 2, (l + r) / 2, r);
		return max(vl, vr);
	}

	ll getMax(int a, int b) {
		// 半開区間に変換
		return getMaxBdy(a, b + 1);
	}

	ll getMaxBdy(int a, int b, int k = 0, int l = 0, int r = -1) {
		if (r < 0) r = N;
		if (b <= l || r <= a) return -big;

		// 関数が呼び出されたら評価！
		eval(k, l, r);
		if (a <= l && r <= b) return node[k];
		ll vl = getMaxBdy(a, b, 2 * k + 1, l, (l + r) / 2);
		ll vr = getMaxBdy(a, b, 2 * k + 2, (l + r) / 2, r);
		return max(vl, vr);
	}
};

class Segment;
class Circle;

class Point {
public:
	double x, y;

	Point(double x = 0, double y = 0) :x(x), y(y) {}

	Point operator + (Point p) { return Point(x + p.x, y + p.y); }
	Point operator - (Point p) { return Point(x - p.x, y - p.y); }
	Point operator * (double a) { return Point(a * x, a * y); }
	Point operator / (double a) { return Point(x / a, y / a); }

	double abs() { return sqrt(norm()); }
	double norm() { return x * x + y * y; }

	bool operator < (const Point& p)const {
		return x != p.x ? x < p.x : y < p.y;
	}
	bool operator == (const Point& p) const {
		return fabs(x - p.x) < EPS && fabs(y - p.y) < EPS;
	}
	// 内積
	static double dot(Point a, Point b) {
		return a.x * b.x + a.y * b.y;
	}
	// 外積
	static double cross(Point a, Point b) {
		return a.x * b.y - a.y * b.x;
	}
	static bool isOrthogonal(Point a, Point b) {
		return EQ(dot(a, b), 0.0);
	}
	static bool isOrthogonal(Point a1, Point a2, Point b1, Point b2) {
		return isOrthogonal(a1 - a2, b1 - b2);
	}
	static bool isOrthogonal(Segment s1, Segment s2);

	static bool isPalallel(Point a, Point b) {
		return EQ(cross(a, b), 0.0);
	}
	static bool isPalallel(Point a1, Point a2, Point b1, Point b2) {
		return isPalallel(a1 - a2, b1 - b2);
	}
	static bool isPalallel(Segment s1, Segment s2);

	static const int COUNTER_CLOCKWISE = 1;
	static const int CLOCKWISE = -1;
	static const int ONLINE_BACK = 2;
	static const int ONLINE_FRONT = -2;
	static const int ON_SEGMENT = 0;
	static int bbw(Point p0, Point p1, Point p2) {
		// 線分はp0とp1でp2がどこにあるかを探る
		Point a = p1 - p0;
		Point b = p2 - p0;
		if (cross(a, b) > EPS) return COUNTER_CLOCKWISE;
		if (cross(a, b) < -EPS) return CLOCKWISE;
		if (dot(a, b) < -EPS) return ONLINE_BACK;
		if (a.norm() < b.norm()) return ONLINE_FRONT;
		return ON_SEGMENT;
	}

	// 交差しているか
	static bool intersect(Point p1, Point p2, Point p3, Point p4) {
		return (bbw(p1, p2, p3) * bbw(p1, p2, p4) <= 0
			&& bbw(p3, p4, p1) * bbw(p3, p4, p2) <= 0);
	}
	static bool intersect(Segment s1, Segment s2);
	static Point project(Segment s, Point p);

	static Point reflect(Segment s, Point p);

	static double getDistance(Point a, Point b) {
		return (a - b).abs();
	}

	static double getDistanceLP(Segment s, Point p);

	static double getDistanceSP(Segment s, Point p);

	static double getDistance(Segment s1, Segment s2);

	static Point getIntersection(Segment s1, Segment s2);

	static pair<Point, Point> crossPoints(Circle c, Segment s);

	static int contains(vector<Point> g, Point p) {
		int n = g.size();
		bool x = false;
		rep(i, n) {
			Point a = g[i] - p, b = g[(i + 1) % n] - p;
			// 線の上に載っているか
			if (std::abs(cross(a, b)) < EPS && dot(a, b) < EPS) return 1;

			// pを基準として上下にあるか
			// または外積が正か?(→にあるか)
			if (a.y > b.y) swap(a, b);
			if (a.y < EPS && EPS < b.y && cross(a, b) > EPS) x = !x;
		}
		return x ? 2 : 0;
	}

	static vector<Point> andrewScan(vector<Point> s) {
		vector<Point> u, l;
		ll si = s.size();
		if (si < 3) return s;
		sort(all(s));
		u.push_back(s[0]);
		u.push_back(s[1]);
		l.push_back(s[si - 1]);
		l.push_back(s[si - 2]);
		for (int i = 2; i < si; i++) {
			for (int _n = u.size(); _n >= 2 && bbw(u[_n - 2], u[_n - 1], s[i]) > CLOCKWISE; _n--) {
				u.pop_back();
			}
			u.push_back(s[i]);
		}

		for (int i = s.size() - 3; i >= 0; i--) {
			for (int _n = l.size(); _n >= 2 && bbw(l[_n - 2], l[_n - 1], s[i]) > CLOCKWISE; _n--) {
				l.pop_back();
			}
			l.push_back(s[i]);
		}

		reverse(all(l));
		for (int i = u.size() - 2; i >= 1; i--)
		{
			l.push_back(u[i]);
		}

		return l;
	}
	void get_cin() {
		cin >> x >> y;
	}

	static Point rotate(double r, Point p) {
		Point ret;
		ret.x = cos(r) * p.x - sin(r) * p.y;
		ret.y = sin(r) * p.x + cos(r) * p.y;
		return ret;
	}

	static double computePerimeter(const vector<Point>& hull) {
		double perimeter = 0.0;
		for (size_t i = 0; i < hull.size(); i++) {
			perimeter += getDistance(hull[i], hull[(i + 1) % hull.size()]);
		}
		return perimeter;
	}
};

class Segment {
public:
	Point p1, p2;
	Segment() {}
	Segment(Point p1, Point p2) :p1(p1), p2(p2) {}
	void get_cin() {
		cin >> p1.x >> p1.y >> p2.x >> p2.y;
	}
	Point p1tp2() {
		return p2 - p1;
	}
	Point p2tp1() {
		return p1 - p2;
	}
	double abs() {
		return (p2 - p1).abs();
	}
	double norm() {
		return (p2 - p1).norm();
	}
};

// 直行
bool Point::isOrthogonal(Segment s1, Segment s2) {
	return EQ(dot(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}

// 平行
bool Point::isPalallel(Segment s1, Segment s2) {
	return EQ(cross(s1.p2 - s1.p1, s2.p2 - s2.p1), 0.0);
}
// 交差しているか
bool Point::intersect(Segment s1, Segment s2) {
	return intersect(s1.p1, s1.p2, s2.p1, s2.p2);
}
Point Point::project(Segment s, Point p) {
	Point base = s.p2 - s.p1;
	double r = Point::dot(p - s.p1, base) / base.norm();
	return s.p1 + base * r;
}
Point Point::reflect(Segment s, Point p) {
	return (project(s, p) * 2) - p;
}
double Point::getDistanceLP(Segment s, Point p) {
	return std::abs(cross(s.p2 - s.p1, p - s.p1) / (s.p2 - s.p1).abs());
}
double Point::getDistanceSP(Segment s, Point p) {
	if (dot(s.p2 - s.p1, p - s.p1) < 0.0) return (p - s.p1).abs();
	if (dot(s.p1 - s.p2, p - s.p2) < 0.0) return (p - s.p2).abs();
	return getDistanceLP(s, p);
}
double Point::getDistance(Segment s1, Segment s2) {
	if (intersect(s1, s2)) return 0.0;
	return min({ getDistanceSP(s1,s2.p1),getDistanceSP(s1,s2.p2)
		,getDistanceSP(s2,s1.p1),getDistanceSP(s2,s1.p2) });
}

Point Point::getIntersection(Segment s1, Segment s2) {
	// (s1.p1 - s2.p1).norm()
	auto bs = s1.p2 - s1.p1;
	auto n1 = s2.p1 - s1.p1;
	auto n2 = s2.p2 - s1.p1;
	auto c1 = std::abs(cross(n1, bs)) / bs.norm();
	auto c2 = std::abs(cross(n2, bs)) / bs.norm();
	return s2.p1 + (s2.p2 - s2.p1) * (c1 / (c1 + c2));
	// c1:c2=t:1-t
	// c2t=(1-t)c1
	// t/(1-t)=c1/(c1+c2)
	//
}

double arg(Point p) { return atan2(p.y, p.x); }
Point polar(double a, double r) { return Point(cos(r) * a, sin(r) * a); }
class Circle {
public:
	Point c;
	double r;
	Circle(Point c = Point(), double r = 0.0) : c(c), r(r) {}
	void get_cin() {
		cin >> c.x >> c.y >> r;
	}
	static pair<Point, Point> getCrossPoints(Circle c1, Circle c2) {
		double d = (c1.c - c2.c).abs(); // 中心点どうしの距離
		double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
		double t = arg(c2.c - c1.c);
		return make_pair(c1.c + polar(c1.r, t + a), c1.c + polar(c1.r, t - a));
	}
};

pair<Point, Point> Point::crossPoints(Circle c, Segment s) {
	auto pp = project(s, c.c);
	auto f = (pp - c.c).norm();
	auto mu = sqrt(c.r * c.r - f);

	// 単位ベクトル
	auto e = s.p1tp2() / s.p1tp2().abs();
	return make_pair(pp + e * mu, pp - e * mu);
}

ll divRm(string s, ll x) {
	ll r = 0;
	for (ll i = 0; i < s.size(); i++)
	{
		r *= 10;
		r += s[i] - '0';
		r %= x;
	}
	return r;
}
ll cmbi(ll x, ll b) {
	ll res = 1;
	for (size_t i = 0; i < b; i++)
	{
		res *= x - i;
		res %= INF;
		res *= inv[b - i];
		res %= INF;
	}
	return res;
}
map<ll, ll> dgmemo;
ll digsum(ll x) {
	if (dgmemo.count(x))return dgmemo[x];
	ll res = 0;
	while (x > 0)
	{
		res += x % 10;
		x /= 10;
	}
	return res;
}
bool check_parindrome(string s) {
	int n = s.size();
	rep(i, n / 2) {
		if (s[i] != s[n - i - 1]) {
			return false;
		}
	}
	return true;
}
ll npr(ll n, ll r) {
	if (r == 0)
		return 1;
	return inff(fac[n] * modinv(fac[n - r]));
}

vl zalgo(string s) {
	ll c = 0;
	vl a(s.size());
	ll si = s.size();
	rep2(i, 1, s.size()) {
		if (i + a[i - c] < c + a[c])
		{
			a[i] = a[i - c];
		}
		else {
			ll j = max(0LL, a[c] - (i - c));
			while (i + j < si && s[j] == s[i + j])
			{
				j++;
			}

			a[i] = j;
			c = i;
		}
	}
	a[0] = s.size();
	return a;
}
// 数値文字列の除算
string divStrNum(string s, ll v) {
	ll si = s.size();
	ll val = 0;
	string res = "";
	for (ll i = 0; i < si; i++)
	{
		val *= 10;
		val += s[i] - '0';
		ll add = val / v;
		val %= v;
		if (add == 0 && res == "")
			continue;
		res += add + '0';
	}
	if (res == "")
		return "0";

	return res;
}

// 数値文字列の減算
string difStrNum(string s, ll v) {
	ll si = s.size();
	bool dec = false;
	for (ll i = si - 1; i >= 0; i--)
	{
		if (v == 0)
			break;
		ll t = v % 10;
		v /= 10;
		ll u = (s[i] - '0');
		if (dec) {
			if (u == 0) {
				s[i] = 9 - t;
				dec = true;
				continue;
			}
			u--;
		}
		if (u < t) {
			s[i] = 10 - (t - u);
			dec = true;
		}
		else {
			s[i] -= t;
			dec = false;
		}
	}
	return s;
}
// 数値文字列を1減らした数
string decStrNum(string s) {
	ll si = s.size();
	for (int i = si - 1; i >= 0; i--)
	{
		if (s[i] == '0') {
			s[i] = '9';
			continue;
		}
		s[i] = s[i] - 1;

		break;
	}
	return s;
}
void dateCal(int x) {
	int lp = x / 7;
	string date[] = { "月曜日","火曜日","水曜日","木曜日","金曜日","土曜日","日曜日" };
	rep(i, 7) {
		int st = i;
		rep(j, lp) {
			cout << "\t" << date[i] << x << "-" << st << "\t" << "NULL" << "\t" << x << "\t" << st << "\t" << 0 << endl;
			st += 7;
		}
	}
}
// 行列べき乗計算
mat mul(mat& A, mat& B) {
	ll as = A.size();
	ll bs = B.size();
	mat C(A.size(), vl(B[0].size()));
	rep(i, as) {
		rep(t, bs) {
			ll bz = B[0].size();
			rep(j, bz) {
				C[i][j] = inff(C[i][j] + A[i][t] * B[t][j]);
			}
		}
	}
	return C;
}

mat pow(mat A, ll x) {
	if (A.size() == 0)return A;
	mat B(A.size(), vl(A.size()));
	rep(i, A.size()) {
		B[i][i] = 1;
	}
	while (x > 0)
	{
		if (x & 1)
			B = mul(B, A);
		A = mul(A, A);
		x >>= 1;
	}
	return B;
}

class dinic {
public:
	vve G;

	vl level;
	vl iter;
	dinic(int _n) : dinic(vve(_n + 1)) {
	}
	dinic(vve g) {
		G = g;
		level = vl(g.size());
		iter = vl(g.size());
	}

	void add_edge(ll from, ll to, ll cap) {
		auto e1 = edge();
		auto e2 = edge();

		e1.flowEdge(to, cap, G[to].size());
		G[from].push_back(e1);
		e2.flowEdge(from, 0, G[from].size() - 1);
		G[to].push_back(e2);
	}

	void bfs(ll s) {
		fill(all(level), -1);
		queue<ll> q;
		level[s] = 0;
		q.push(s);
		while (!q.empty())
		{
			ll v = frontpop(q);
			for (auto e : G[v]) {
				if (e.cap > 0 && level[e.to] < 0) {
					level[e.to] = level[v] + 1;
					q.push(e.to);
				}
			}
		}
	}

	ll dfs(ll v, ll t, ll f) {
		if (v == t)
			return f;
		for (ll& i = iter[v]; i < G[v].size(); i++) {
			edge& e = G[v][i];
			if (e.cap > 0 && level[v] < level[e.to]) {
				ll d = dfs(e.to, t, min(f, e.cap));
				if (d > 0) {
					e.cap -= d;
					G[e.to][e.rev].cap += d;
					return d;
				}
			}
		}
		return 0;
	}

	ll max_flow(ll s, ll t) {
		ll flow = 0;
		for (;;) {
			bfs(s);
			if (level[t] < 0)
				return flow;
			fill(all(iter), 0);
			ll f;
			while ((f = dfs(s, t, big)) > 0)
			{
				flow += f;
			}
		}
	}
};
const ull BS = 1000000007;
// aはbに含まれているか？
bool rolling_hash(string a, string b) {
	int al = a.size(), bl = b.size();
	if (al > bl)
		return false;

	// BSのal乗を計算
	ull t = 1;
	rep(i, al)t *= BS;

	// aとbの最初のal文字に関するハッシュ値を計算
	ull ah = 0, bh = 0;
	rep(i, al) ah = ah * BS + a[i];
	rep(i, al) bh = bh * BS + b[i];

	// bの場所を一つずつ進めながらハッシュ値をチェック
	for (ll i = 0; i + al <= bl; i++)
	{
		if (ah == bh)
			return true;
		if (i + al < bl)bh = bh * BS + b[i + al] - b[i] * t;
	}
	return false;
}

mat sans(9, vl(9, -1));
bool srec(ll x, ll y) {
	if (x == 9)
		return true;
	vl use(10, 0);
	rep(i, 9) {
		if (sans[i][y] == -1)
			continue;
		use[sans[i][y]] = 1;
	}
	rep(j, 9) {
		if (sans[x][j] == -1)
			continue;
		use[sans[x][j]] = 1;
	}
	ll px = x % 3;
	ll py = y % 3;
	ll tx = x - px + 3;
	ll ty = y - py + 3;
	rep2(i, x - px, tx) {
		rep2(j, y - py, ty) {
			if (sans[i][j] == -1)
				continue;
			use[sans[i][j]] = 1;
		}
	}
	ll nx, ny;
	if (y == 8) {
		nx = x + 1;
		ny = 0;
	}
	else {
		nx = x;
		ny = y + 1;
	}

	if (sans[x][y] != -1) {
		if (srec(nx, ny)) {
			return true;
		}
		return false;
	}

	rep2(i, 1, 10) {
		if (use[i])
			continue;
		sans[x][y] = i;
		if (srec(nx, ny)) {
			return true;
		}
		sans[x][y] = -1;
	}
	return false;
}
void sudoku() {
	vector<string> tb;

	rep(i, 9) {
		string s;
		cin >> s;
		tb.push_back(s);
		rep(j, 9) {
			if (tb[i][j] != '.') {
				sans[i][j] = tb[i][j] - '0';
			}
		}
	}
	srec(0, 0);
	rep(i, 9) {
		rep(j, 9) {
			cout << sans[i][j];
		}
		cout << endl;
	}
}
mint ncr(ll n, ll  r) {
	mint v = 1;
	rep(i, r) {
		v *= (n - i);
		v *= inv[i + 1];
	}
	return v;
}
modint1000000007 ncr2(ll n, ll r) {
	modint1000000007 v = 1;
	rep(i, r) {
		v *= (n - i);
		v /= i + 1;
	}
	return v;
}

ll sq(ll x) {
	return x * x;
}
ll phi(ll x) {
	auto p = getp(x);
	ll res = x;
	for (auto v : p) {
		res /= v;
		res *= v - 1;
	}
	return res;
}
const ull MASK30 = (1ULL << 30) - 1;
const ull MASK31 = (1ULL << 31) - 1;
const ull MOD = 2305843009213693953UL;
const ull MASK61 = (1ULL << 61UL) - 1UL;
//mod 2^61-1を計算する関数
ull calc_mod_61(ull x)
{
	ull xu = x >> 61;
	ull xd = x & MASK61;
	ull res = xu + xd;
	if (res >= MOD) res -= MOD;
	return res;
}
ull mul_61(ull a, ull b)
{
	ull au = a >> 31;
	ull ad = a & MASK31;
	ull bu = b >> 31;
	ull bd = b & MASK31;
	ull mid = ad * bu + au * bd;
	ull midu = mid >> 30;
	ull midd = mid & MASK30;
	return calc_mod_61(au * bu * 2 + midu + (midd << 31) + ad * bd);
}

vl mulMatVec(mat a, vl b)
{
	int n = b.size(); vl ret(n, 0);
	rep(i, n) rep(j, n)
		ret[j] = inff(ret[j] + inff(a[i][j] * b[i]));
	return ret;
}
ll isqrt(ll N) {
	ll sqrtN = sqrt(N) - 1;
	while (sqrtN + 1 <= N / (sqrtN + 1))sqrtN++;
	return sqrtN;
}
ll cross(pll l, pll r) {
	return l.first * r.second - l.second * r.first;
}
void rotate(vl& v) {
	v.push_back(v.front());
	v.erase(v.begin());
}


class ConvexHullDynamic
{
	typedef long long coef_t;
	typedef long long coord_t;
	typedef long long val_t;

	/*
	* Line 'y=a*x+b' represented by 2 coefficients 'a' and 'b'
	* and 'xLeft' which is intersection with previous line in hull(first line has -INF)
	*/
private:
	struct Line
	{
		coef_t a, b;
		double xLeft;

		enum Type
		{
			line, maxQuery, minQuery
		} type;
		coord_t val;

		explicit Line(coef_t aa = 0, coef_t bb = 0) : a(aa), b(bb), xLeft(-INFINITY), type(Type::line), val(0) {}

		val_t valueAt(coord_t x) const { return a * x + b; }

		friend bool areParallel(const Line& l1, const Line& l2) { return l1.a == l2.a; }

		friend double intersectX(const Line& l1, const Line& l2) { return areParallel(l1, l2) ? INFINITY : 1.0 * (l2.b - l1.b) / (l1.a - l2.a); }

		bool operator<(const Line& l2) const
		{
			if (this->type == maxQuery)
				return this->val < l2.xLeft;
			if (this->type == minQuery)
				return this->val > l2.xLeft;
			if (l2.type == line)
				return this->a < l2.a;
			if (l2.type == maxQuery)
				return this->xLeft < l2.val;
			if (l2.type == minQuery)
				return this->xLeft > l2.val;
		}
	};


	bool isMax; //whether or not saved envelope is top(search of max value)
public:
	std::set< Line > hull;  //envelope itself

private:
	/*
	* INFO:        Check position in hull by iterator
	* COMPLEXITY:  O(1)
	*/
	bool hasPrev(std::set< Line >::iterator it) { return it != hull.begin(); }

	bool hasNext(std::set< Line >::iterator it) { return it != hull.end() && std::next(it) != hull.end(); }

	/*
	* INFO:        Check whether line l2 is irrelevant
	* NOTE:        Following positioning in hull must be true
	*              l1 is next left to l2
	*              l2 is right between l1 and l3
	*              l3 is next right to l2
	* COMPLEXITY:  O(1)
	*/
	bool irrelevant(const Line& l1, const Line& l2, const Line& l3) { return intersectX(l1, l3) <= intersectX(l1, l2); }

	bool irrelevant(std::set< Line >::iterator it)
	{
		return hasPrev(it) && hasNext(it)
			&& (isMax && irrelevant(*std::prev(it), *it, *std::next(it))
				|| !isMax && irrelevant(*std::next(it), *it, *std::prev(it)));
	}

	/*
	* INFO:        Updates 'xValue' of line pointed by iterator 'it'
	* COMPLEXITY:  O(1)
	*/
	std::set< Line >::iterator updateLeftBorder(std::set< Line >::iterator it)
	{
		if (isMax && !hasPrev(it) || !isMax && !hasNext(it))
			return it;

		double val = intersectX(*it, isMax ? *std::prev(it) : *std::next(it));
		Line buf(*it);
		it = hull.erase(it);
		buf.xLeft = val;
		it = hull.insert(it, buf);
		return it;
	}

public:
	explicit ConvexHullDynamic(bool isMax = false) : isMax(isMax) {}

	/*
	* INFO:        Adding line to the envelope
	*              Line is of type 'y=a*x+b' represented by 2 coefficients 'a' and 'b'
	* COMPLEXITY:  Adding N lines(N calls of function) takes O(N*log N) time
	*/
	void addLine(coef_t a, coef_t b)
	{
		//find the place where line will be inserted in set
		Line l3 = Line(a, b);
		auto it = hull.lower_bound(l3);

		//if parallel line is already in set, one of them becomes irrelevant
		if (it != hull.end() && areParallel(*it, l3)) {
			if (isMax && it->b < b || !isMax && it->b > b)
				it = hull.erase(it);
			else
				return;
		}

		//try to insert
		it = hull.insert(it, l3);
		if (irrelevant(it)) {
			hull.erase(it);
			return;
		}

		//remove lines which became irrelevant after inserting line
		while (hasPrev(it) && irrelevant(std::prev(it))) hull.erase(std::prev(it));
		while (hasNext(it) && irrelevant(std::next(it))) hull.erase(std::next(it));

		//refresh 'xLine'
		it = updateLeftBorder(it);
		if (hasPrev(it))
			updateLeftBorder(std::prev(it));
		if (hasNext(it))
			updateLeftBorder(std::next(it));
	}

	val_t getBest(coord_t x) const
	{
		Line q;
		q.val = x;
		q.type = isMax ? Line::Type::maxQuery : Line::Type::minQuery;

		auto bestLine = hull.lower_bound(q);
		if (isMax) --bestLine;
		return bestLine->valueAt(x);
	}


};
class treelib {

public:
	mat es;
	vl stop;
	vl d;
	treelib(mat _es) : es(_es) {
		stop.resize(_es.size() + 1, 0);
		d.resize(_es.size() + 1);
	}

	/*
	* first: depth.second : leaf;
	*/
	pll deepest(ll x, ll f) {
		ll a = 0, b = -1;
		for (auto v : es[x]) {
			if (stop[v])continue;
			if (v == f)continue;
			d[v] = d[x] + 1;
			auto p = deepest(v, x);
			if (p.first > a) {
				a = p.first;
				b = p.second;
			}
		}
		if (b == -1) {
			return { 1,x };
		}
		else {
			return { a + 1,b };
		}
	}
};

class treelib2 {

public:
	vector<vpll> es;
	vl stop;
	vl d;
	treelib2(vector<vpll> _es) : es(_es) {
		stop.resize(_es.size() + 1, 0);
		d.resize(_es.size() + 1);
	}

	/*
	* first: depth.second : leaf;
	*/
	pll deepest(ll x, ll f) {
		ll a = 0, b = -1;
		for (auto v : es[x]) {
			ll t = v.first;
			if (stop[t])continue;
			if (t == f)continue;
			d[t] = d[x] + v.second;
			auto p = deepest(t, x);
			if (p.first > a) {
				a = p.first;
				b = p.second;
			}
		}
		if (b == -1) {
			return { 1,x };
		}
		else {
			return { a + 1,b };
		}
	}
};
struct scompress {
	vl mapped, dup;
	map<ll, ll> mp;
};
scompress compress(vl& v) {
	ll n = v.size();
	vl b(n);
	rep(i, n) {
		b[i] = v[i];
	}
	vsort(b);
	dup(b);
	map<ll, ll> mp;
	rep(i, b.size()) {
		mp[b[i]] = i;
	}
	vl res(n);
	rep(i, n) {
		res[i] = mp[v[i]];
	}
	vl bb(b.size());
	rep(i, b.size()) {
		bb[i] = mp[b[i]];
	}
	return { res,bb,mp };
}
using ld = double;
using P = Point;
template <class iter>
Circle min_ball(iter left, iter right, int seed = 1333) {
	const int n = right - left;

	assert(n >= 1);
	if (n == 1) {
		return { *left, ld(0) };
	}

	std::mt19937 mt(seed);
	std::shuffle(left, right, mt);
	// std::random_shuffle(left, right); // simple but deprecated

	iter ps = left;
	using circle = Circle;

	auto make_circle_3 = [](P& a, P& b, P& c) -> circle {
		ld A = (b - c).norm(), B = (c - a).norm(), C = (a - b).norm(),
			S = Point::cross(b - a, c - a);
		P p = (a * (A * (B + C - A)) + (b * B * (C + A - B)) + c * C * (A + B - C))
			/ (4 * S * S);
		ld r2 = (p - a).norm();
		return { p, r2 };
		};

	auto make_circle_2 = [](P& a, P& b) -> circle {
		P c = (a + b) / (ld)2;
		ld r2 = (a - c).norm();
		return { c, r2 };
		};

	auto in_circle = [](P& a, circle& c) -> bool {
		return (a - c.c).norm() <= c.r + EPS;
		};

	circle c = make_circle_2(ps[0], ps[1]);

	// MiniDisc
	for (int i = 2; i < n; ++i) {
		if (!in_circle(ps[i], c)) {
			// MiniDiscWithPoint
			c = make_circle_2(ps[0], ps[i]);
			for (int j = 1; j < i; ++j) {
				if (!in_circle(ps[j], c)) {
					// MiniDiscWith2Points
					c = make_circle_2(ps[i], ps[j]);
					for (int k = 0; k < j; ++k) {
						if (!in_circle(ps[k], c)) {
							c = make_circle_3(ps[i], ps[j], ps[k]);
						}
					}
				}
			}
		}
	}
	return c;
}
vml2 kitamasadfs(vml2 a, vml2 d, ll n) {
	if (d.size() == n)
		return d;
	vml2 res(d.size());
	if (n < d.size() * 2 || (n & 1)) {
		auto f = kitamasadfs(a, d, n - 1);
		res[0] = f[k - 1] * d[0];
		rep2(i, 1, d.size()) {
			res[i] = f[i - 1] + f[k - 1] * d[i];
		}
	}
	else {
		auto v = kitamasadfs(a, d, n / 2);
		matm2 f(d.size(), vml2(d.size()));
		f[0] = v;
		rep2(i, 1, d.size()) {
			f[i][0] = f[i - 1][k - 1] * d[0];
			rep2(j, 1, d.size()) {
				f[i][j] = f[i - 1][j - 1] + f[i - 1][k - 1] * d[j];
			}
		}

		rep(i, d.size()) {
			rep(j, d.size()) {
				res[j] += f[i][j] * v[i];
			}
		}
	}

	return res;
}
modint1000000007 kitamasa(vml2  a, vml2 d, ll n) {
	auto v = kitamasadfs(a, d, n);
	modint1000000007 res = 0;
	rep(i, d.size()) {
		res += v[i] * a[i];
	}
	return res;
}
void belman_temp(vector<vpll>& es, vl& d, ll s) {
	d[s] = 0;
	rep(i, n + 1) {
		queue<ll> q;
		rep2(j, 1, n + 1) {
			if (d[j] == big)continue;
			for (auto& v : es[j]) {
				if (chmin(d[v.first], d[j] + v.second)) {
					q.push(v.first);
				}
			}
		}
		if (i < n)continue;
		while (!q.empty())
		{
			auto p = frontpop(q);
			for (auto& v : es[p]) {
				if (chmin(d[v.first], -big)) {
					q.push(v.first);
				}
			}
		}
	}
}
vl getpath(mat& es, vl& d, ll s, ll g) {
	vl res;
	ll x = s;
	while (x != g)
	{
		res.push_back(x);
		for (auto v : es[x]) {
			if (d[v] == d[x] - 1) {
				x = v;
				break;
			}
		}
	}
	res.push_back(x);
	reverse(all(res));
	return res;
}
/// <summary>
/// ベルマンフォード
/// </summary>
/// <param name="es"></param>
/// <param name="d"></param>
/// <param name="s"></param>
bool belman(vector<vpll>& es, ll n, vl& d, ll s) {

	d.resize(n, big);
	d[s] = 0;
	rep(i, n) {
		bool e = false;
		rep(f, n) {
			if (d[f] == big)continue;
			for (auto& v : es[f]) {
				if (chmin(d[v.first], d[f] + v.second)) {
					e = true;
				}
			}
		}
		if (!e) break;
	}

	queue<ll> q;
	rep(f, n) {
		if (d[f] == big)continue;
		for (auto& v : es[f]) {
			if (chmin(d[v.first], d[f] + v.second)) {
				q.push(v.first);
			}
		}
	}
	bool e = false;
	while (!q.empty())
	{
		auto p = frontpop(q);
		for (auto& v : es[p]) {
			if (d[v.first] > -big) {
				e = true;
				d[v.first] = -big;
				q.push(v.first);
			}
		}
	}
	return e;
}
template<class t>
void put_line(vector<t>& p) {
	rep(i, p.size()) {
		cout << p[i] << " ";
	}
	cout << endl;
}

mat tablecut(ll h, ll w, vector<string> t) {
	ll top = 0;
	rep(i, h) {
		bool ok = true;

		rep(j, w) {
			if (t[i][j] == '#') {
				ok = false;
				break;
			}
		}
		if (!ok)break;
		top++;
	}
	ll bot = h;
	for (int i = h - 1; i >= 0; i--)
	{
		bool ok = true;

		rep(j, w) {
			if (t[i][j] == '#') {
				ok = false;
				break;
			}
		}
		if (!ok)break;
		bot--;
	}

	ll lf = 0;
	rep(i, w) {
		bool ok = true;
		rep(j, h) {
			if (t[j][i] == '#') {
				ok = false;
				break;
			}
		}
		if (!ok)break;
		lf++;;
	}
	ll ri = w;
	for (int i = w - 1; i >= 0; i--)
	{
		bool ok = true;
		rep(j, h) {
			if (t[j][i] == '#') {
				ok = false;
				break;
			}
		}
		if (!ok)break;
		ri--;
	}

	mat tb(bot - top, vl(ri - lf));
	rep2(i, top, bot) {
		rep2(j, lf, ri) {
			if (t[i][j] == '#') {
				tb[i - top][j - lf] = 1;
			}
		}
	}
	return tb;
}

mat tablerotate(ll h, ll w, mat& a) {
	mat b(w, vl(h));
	rep(i, h) {
		rep(j, w) {
			b[w - j - 1][i] = a[i][j];
		}
	}
	return b;
}


ll rangeadd_op(ll l, ll r) {
	return max(l, r);
}
ll rangeadd_e() {
	return -big;
}

ll range_add_map(ll l, ll r) {
	if (l == -big)return r;
	if (r == -big)return l;
	return l + r;
}
ll range_add_comp(ll l, ll r) {
	return l + r;
}
ll rangeadd_id() {
	return 0;
}
ll rangesum_op(ll l, ll r) {
	return max(0LL, l) + max(0LL, r);
}
ll rsum_op(ll l, ll r) {
	return l + r;
}
ll rangesum_e() {
	return -big;
}

struct Qusm {
	ll a = 0, sz = 0;
};
Qusm opQusm(Qusm l, Qusm r) {
	return { l.a + r.a,l.sz + r.sz };
}
Qusm eQusm() {
	Qusm q;
	return q;
}
Qusm mapQusm(ll l, Qusm v) {
	return { v.a + v.sz * l,v.sz };
}
ll cmpQusm(ll ne, ll ol) {
	return ne + ol;
}
ll idQusm() {
	return 0;
}
lazy_segtree<ll, rangeadd_op, rangeadd_e, ll, range_add_map, range_add_comp, rangeadd_id>
create_range_add_st(ll n) {
	return lazy_segtree<ll,
		rangeadd_op,
		rangeadd_e,
		ll,
		range_add_map,
		range_add_comp,
		rangeadd_id>(n + 1);
}
lazy_segtree<Qusm, opQusm, eQusm, ll, mapQusm, cmpQusm, idQusm>
create_range_add_st3(ll n) {
	return lazy_segtree<Qusm, opQusm, eQusm, ll, mapQusm, cmpQusm, idQusm>(n + 1);
}

lazy_segtree<ll, rangesum_op, rangesum_e, ll, range_add_map, range_add_comp, rangeadd_id>
create_range_add_stv2(vl a) {
	return lazy_segtree<ll,
		rangesum_op,
		rangesum_e,
		ll,
		range_add_map,
		range_add_comp,
		rangeadd_id>(a);
}

class rolhash_lib {
	string s;
	vl v, p;
	ll n;
public:
	rolhash_lib() {

	}
	rolhash_lib(string _s) : s(_s) {
		n = s.size();
		v.resize(n + 1);
		p.resize(n + 1);
		p[0] = 1;
		rep(i, n) {
			v[i + 1] = calc_mod_61(mul_61(v[i], INF) + s[i]);
			p[i + 1] = mul_61(p[i], INF);
		}
	}

	ll get_hash(ll l, ll r) {
		l--;
		return calc_mod_61(v[r] + MOD * 4 - mul_61(v[l], p[r - l]));
	}
};

template<class t>
class zobhash_lib {
	vector<t> s;
	vul v, p;
	ll n;
public:
	zobhash_lib() {

	}
	zobhash_lib(vector<t> _s, vector<t> vals) : s(_s) {
		n = s.size();
		v.resize(n + 1);
		p.resize(n + 1);
		p[0] = 1;
		map<t, ull> mp;
		ull q = INF;
		rep(i, vals.size()) {
			mp[vals[i]] = mul_61(vals[i], q);
			q = mul_61(q, INF);
		}
		rep(i, n) {
			v[i + 1] = calc_mod_61(v[i] + mp[s[i]]);
		}
	}

	ull get_hash(ll l, ll r) {
		l--;
		return calc_mod_61(v[r] + MOD * 4 - v[l]);
	}
};

long long llceil(long long a, long long b) {
	if (a % b == 0) { return a / b; }

	if (a >= 0) { return (a / b) + 1; }
	else { return -((-a) / b); }
}

long long llfloor(long long a, long long b) {
	if (a % b == 0) { return a / b; }

	if (a >= 0) { return (a / b); }
	else { return -((-a) / b) - 1; }
}

using pl = pair<long long, long long>;
pl findseg(pl seg, long long ini, long long step) {
	if (step > 0) {
		return { llceil(seg.first - ini,step), llfloor(seg.second - ini,step) };
	}
	else {
		step *= -1;
		return { llceil(ini - seg.second,step), llfloor(ini - seg.first,step) };
	}
}

ll matsum(mat& a, ll i, ll j, ll x, ll y) {
	return a[i][j] - a[i - x][j] - a[i][j - y] + a[i - x][j - y];
}
ll __parity(ll t) {
	ll c = 0;
	while (t > 0)
	{
		c += t & 1;
		t >>= 1;
	}
	return c % 2;
}
ll lcm(ll a, ll b) {
	return a * b / gcd(a, b);
}

ll popcount(ll x) {
	ll c = 0;
	while (x > 0)
	{
		c += x & 1;
		x >>= 1;
	}
	return c;
}

struct centroid_decomposition {
	int n;
	int centor;
	mat G;
	vector<int>size;
	vector<vector<array<ll, 3>>>child; //child[i]=iが重心の木の、iを根としたときの子の(index,size,centoroid index)
	vector<bool>removed; //作業用
	centroid_decomposition(mat& g) {
		G = g;
		n = G.size();
		size.resize(n);
		child.resize(n);
		removed.resize(n);
		decompose();
	};

	int find_centroid(int v, int pre, int cnt) {
		// 残っている頂点でなす、vを含む連結成分における重心のindexを返す
		// early returnはせず、sizeの再計算を全部やる
		size[v] = 1;
		bool ok = true;
		int centor = -1;
		for (auto vv : G[v]) {
			if (vv == pre)continue;
			if (removed[vv])continue;
			centor = max(centor, find_centroid(vv, v, cnt));

			size[v] += size[vv];
			ok &= size[vv] <= cnt / 2;
		}
		ok &= cnt - size[v] <= cnt / 2;
		return ok ? v : centor;
	}

	int decompose_recursive(int v, int cnt) {
		int vv = find_centroid(v, -1, cnt);
		removed[vv] = true;
		for (auto vvv : G[vv])if (!removed[vvv]) {
			int bbc = size[vvv] < size[vv] ? size[vvv] : cnt - size[vv];
			child[vv].push_back({ vvv,bbc,-1 });
		}
		for (auto& item : child[vv])item[2] = decompose_recursive(item[0], item[1]);
		return vv;
	}

	void decompose() {
		centor = decompose_recursive(0, n);
	}
};
template <typename T>
vl argsort(const vector<T>& A) {
	// stable
	vl ids(A.size());
	iota(all(ids), 0);
	sort(all(ids),
		[&](int i, int j) { return A[i] < A[j] || (A[i] == A[j] && i < j); });
	return ids;
}
// A[I[0]], A[I[1]], ...
template <typename T>
vector<T> rearrange(const vector<T>& A, const vl& I) {
	int n = A.size();
	vector<T> B(n);
	rep(i, n) B[i] = A[I[i]];
	return B;
}

bool intersection(ll f, ll t, ll ff, ll tt) {
	return !(tt <= f || t <= ff);
}

//　ここまでライブラリ
// ここからコード
struct C {
	ll a, mi;
};
struct O {
	ll l, r, q;
};
struct S {
	ll sz, val;
};
S op(S l, S r) {
	return { l.sz + r.sz,l.val + r.val };
}

S e() {
	return { 0,0 };
}
ll ore() {
	return 0;
}
S mapping(ll f, S s) {
	if (f == -1)return s;
	return { s.sz,f * s.sz };
}
ll mapma(ll v, ll x) {
	if (v < 0)return x;
	return v;
}
ll composition(ll ne, ll ol) {
	if (ne < 0)return ol;
	if (ol < 0)return ne;
	return ne;
}
ll id() {
	return -1;
}

ll opmin(ll l, ll r) {
	return min(l, r);
}
ll emin() {
	return big;
}

ll opma(ll l, ll r) {
	return max(l, r);
}
ll ema() {
	return -big;
}
ll mamapping(ll ne, ll o) {
	if (ne < 0)return o;
	return ne;
}
ll changeCom(ll ne, ll ol) {
	if (ne < 0)return ol;
	return ne;
}
ll changeee() {
	return -1;
}
ll oppp(ll l, ll r) {
	return max(l, r);
}

ll ee() {
	return -big;
}

modint998244353 o1(modint998244353 l, modint998244353 r) {
	return l + r;
}
modint998244353 e1() {
	return 0;
}

struct F {
	ll lz = 0, lo = 0, rz = 0, ro = 0, mz = 0, mo = 0, len = 0;
};
F ost(F l, F r) {
	if (l.len == -1)return r;
	if (r.len == -1)return l;
	ll lz = l.lz;
	ll lo = l.lo;
	ll rz = r.rz;
	ll ro = r.ro;
	if (rz == r.len) {
		rz += l.rz;
	}
	if (ro == r.len) {
		ro += l.ro;
	}
	if (lz == l.len) {
		lz += r.lz;
	}
	if (lo == l.len) {
		lo += r.lo;
	}
	ll sm = l.len + r.len;
	ll mo = max({ l.mo	,r.mo,l.ro + r.lo });
	ll mz = max({ l.mz,r.mz, l.rz + r.lz });
	return { lz,lo,rz,ro,mz,mo,sm };
}

F est() {
	return { -1,-1,-1,-1,-1,-1,-1 };
}
F maest(ll v, F s) {
	if (v % 2 == 0)return s;
	return { s.lo,s.lz,s.ro,s.rz,s.mo,s.mz,s.len };
}
vl o157(vl l, vl r) {
	if (l.empty())return r;
	if (r.empty())return l;
	rep(i, 26) {
		r[i] += l[i];
	}
	return r;
}
vl e157() {
	return {};
}
double ops(double l, double r) {
	return l + r;
}
double ope() {
	return 0;
}
pair<vl, vl> opx(pair<vl, vl> l, pair<vl, vl> r) {
	if (l.first.empty())return r;
	if (r.first.empty())return l;

	vl cn(26), tn(26);
	for (int i = 25; i >= 0; i--)
	{
		cn[i] = l.first[i];
		if (i < 25) {
			cn[i] += cn[i + 1];
			if (r.first[i] > 0)
				r.second[i] += cn[i + 1];
		}
		r.second[i] += l.second[i];
		r.first[i] += l.first[i];
	}

	return r;
}
pair<vl, vl> epx() {
	return { {},{} };
}
char cnt[162000001];
pll op299(pll l, pll r) {
	if (l.first == -1)return r;
	if (r.first == -1)return l;

	if (l.first < r.first)return l;
	if (l.first > r.first)return r;
	if (l.second < r.second)return l;
	return r;
}
pll e299() {
	return { -1,-1 };
}

pair<ull, ull> oprol(pair<ull, ull> l, pair<ull, ull> r) {

	pair<ull, ull> nx;
	nx.first = calc_mod_61(l.first + mul_61(r.first, l.second));
	nx.second = mul_61(l.second, r.second);
	return nx;
}
pair<ull, ull> erol() {
	return { 0,1 };
}

class rolhash_lib_2 {
	string s;
	vl v, p;
	ll n;
public:
	//pair<ull, ull> op(pair<ull,ull> l, pair<ull, ull> r) {
	//	if (l.first == 0)return r;
	//	if (r.first == 0)return l;
	//	pair<ull,ull> nx;
	//	nx.first = calc_mod_61(mul_61(l.first, INF) + r.first);
	//	nx.second = mul_61(mul_61(l.second, r.second), INF);
	//	return nx;
	//}
	//pair<ull, ull> e() {
	//	return { 0,0 };
	//}

	segtree<pair<ull, ull>, oprol, erol> st;
	rolhash_lib_2(string _s) : s(_s) {
		n = s.size();
		//v.resize(n + 1);
		//p.resize(n + 1);
		//p[0] = 1;
		vector<pair<ull, ull>> v(n);
		rep(i, n) {
			v[i].first = s[i];
			v[i].second = INF;
			//v[i + 1] = calc_mod_61(mul_61(v[i], INF) + s[i]);
			//p[i + 1] = mul_61(p[i], INF);
		}
		st = segtree<pair<ull, ull>, oprol, erol>(v);
	}
	void set(ll x, char c) {
		st.set(x, { c,INF });
	}
	ull get_hash(ll l, ll r) {
		l--;
		return st.prod(l, r).first;
		//return calc_mod_61(v[r] + MOD * 4 - mul_61(v[l], p[r - l]));
	}
};

struct Qu {
	mint9 a = 1, b = 0;
};

ll mapQu(Qu q, ll v) {
	return (v * q.a + q.b).val();
}
Qu cmpQu(Qu ne, Qu ol) {
	return {
		ol.a * ne.a,
		ol.b * ne.a + ne.b
	};
}
Qu idQu() {
	Qu q; return q;
}

struct Quu {
	ll a = 0, ba = 0, b = 0, i = -1;
};
Quu opQuu(Quu l, Quu r) {
	if (r.i == -1) {
		return l;
	}
	if (l.i == -1) {
		return r;
	}
	return { max(l.a + r.a,l.ba + r.a),max({l.ba,l.a}) * r.b + r.ba,l.b * r.b,max(l.i,r.i) };
}
Quu eQuu() {
	Quu q;
	return q;
}
ll se() {
	return 0;
}
void solv() {
	/*
		私は素因数分解を使うべきところで、エラトステネスを使ってハマりました。
		私は「lからrまでを数としてみた時、7で割り切れるか？」を「lからrまでを数としてみた時、『各桁の和を』7で割り切れるか？」と誤解しました。
		私は累積和を使うべきところで、遅延セグ木を使ってTLEを食らいました。
		tをn進法にする時は素直にwhile(t>0)の条件で処理しよう
		問題を誤読すると痛いよ！
		愚直解テストはレンジの小さい範囲も入念に試しておきたい(https://atcoder.jp/contests/abc309/tasks/abc309_f)
		next_permutation使う時は基本的にはソートするんや
		m回接続(ループ)してその中を計算するタイプの問題、確定している分はしっかりmから引く事
		ARCでは特に、愚直解との比較で間違っている箇所は出来る限り出す
		中央値を使う総和の計算の左側は、カッコを忘れない事→x*lf-(s[i]-s[i-lf])
		lazy_segtreeは分解した式で考える
		double の値を10^x倍して小数点四捨五入するときはroundlを使う
	*/

	auto calc = [&]()->ll {
		ll n, m;
		cin >> n >> m;
		vl l(n), r(n);
		ll ls = 0,rs=0;
		rep(i, n) {
			cin >> l[i];
			ls += l[i];
		}
		rep(i, n) {
			cin >> r[i];
			rs += r[i];
		}
		if (m < ls || rs < m) {
			return -1;
		}
		if (ls == m) {
			ll sm = 0;
			rep(i, n) {
				sm += sq(l[i]);

			}
			return (m * m - sm) / 2;
		}
		auto cal = [&](ll x)->pll {
			ll s = 0;
			rep(i, n) {
				ll t = clamp(x, l[i], r[i]);
				s += t;
			}
			return { s <= m,s };
			};

		ll hi = big;
		ll lo = 0;
		while (lo+1<hi)
		{
			ll md = (hi + lo) / 2;
			if (cal(md).first) {
				lo = md;
			}
			else {
				hi = md;
			}
		}

		auto d = cal(lo);
		ll t = lo;
		ll rm = m - d.second;
		ll sm = 0;
		rep(i, n) {
			ll x = clamp(t, l[i], r[i]);
			if(t>=l[i] && t<r[i] && rm>0){
				rm--;
				x++;
			}
			sm += sq(x);
		}
		return (m * m - sm) / 2;
		};
	ll t; cin >> t;
	rep(i, t) {
		pln(calc());
	}
}

int main()
{
	cin.tie(0);
	ios::sync_with_stdio(false);
	//INF = 998244353;
	solv();

	return 0;
}