#include<iostream>
#include<algorithm>
#include<string>
#include<cstring>
#include<sstream>
#include<iomanip>
#include<limits>
#include<deque>
#include<map>
#include<list>
#include<set>
#include<unordered_set>
#include<vector>
#include<cmath>
#include<cstdio>
#include<memory>
#include<bitset>
#include<stack>
#include<functional>
#include<queue>
#include<regex>
#include<time.h>
#include<type_traits>
#include<cstdlib>
#include<utility>
#include<numeric>
#include<iterator>
#include<random>

using namespace std;
using ll = long long;
using grid = vector<vector<char>>;

/*
メモ書き
cout << setw(5) << setfill('0') << 〇〇 << endl;
は5つ右寄せで0埋め出力
*/

constexpr ll MOD = 1000000007;//良く出てくるMOD
constexpr ll INF = 1050000000;//intで使うでかい数
constexpr ll LONGINF = 1050000000000000000;//longlongで使うでかい数

struct all_init {

	//初期化のためだけの構造体
	//コンストラクタが呼ばれ、cin,cout高速化がされる
	//ついでに少数も出力できるようにしている
	all_init() {
		cout.tie(nullptr);
		cin.tie(nullptr);
		ios::sync_with_stdio(false);
		cout << fixed << setprecision(15);
	};
}ALL_INIT;
struct edge {
	//辺の重みを管理できるような構造体
	//フローで使う容量を意味するcapaも追加
	//コンストラクタによって簡単に値を入れられるようにしている
	//operatorは辺の重みでソート出来るようにしている(主に最小全域木用)

	int from, to;
	ll cost;
	ll capa;

	edge(int s, int d) : from(s), to(d) {
		cost = 0; capa = 0;
	}
	edge(int s, int d, ll w) : from(s), to(d), cost(w) { capa = 0; }
	edge(int s, int d, ll x, ll y) :from(s), to(d), cost(x), capa(y) {  }

	bool operator < (const edge& x) const {
		return cost < x.cost;
	}
};
using graph = vector<vector<edge>>;

#define CIN(vector_array_etc,n) for(int loop=0;loop<n;loop++){cin>>vector_array_etc[loop];}
#define COUT(vector_array_etc,n) for(int LOOP=0;LOOP<n;LOOP++){cout<<vector_array_etc[LOOP]<<(LOOP == n-1 ?'\n':' ');}
#define VC(Type_name) vector<Type_name>//1次元ならあまり意味ないかも
#define SORT(vector_etc) sort(vector_etc.begin(),vector_etc.end())
#define ALL(vec_etc) vec_etc.begin(),vec_etc.end()
#define VCVC(Type_name) vector<vector<Type_name>>//2次元配列定義怠過ぎ問題
#define WARSHALL vector<vector<ll>> g(n,vector<ll>(n,LONGINF))
#define endl '\n'
#define YES cout<<"YES"<<endl
#define NO cout<<"NO"<<endl
#define Yes cout<<"Yes"<<endl
#define No cout<<"No"<<endl
#define yes cout<<"yes"<<endl
#define no cout<<"no"<<endl
#define Tof(x) x ? Yes : No
#define TOF(x) x ? YES : NO
#define tof(x) x ? yes : no

template<class T>bool chmax(T &a, const T &b) {
	if (a < b) {
		a = b;
		return true;
	}
	return false;
}//aに最大値が入る
template<class T>bool chmin(T &a, const T &b) {
	if (b < a) {
		a = b;
		return true;
	}
	return false;
}//aに最小値が入る
template<typename T>istream& operator >> (istream& is, vector<T>& Vec) {
	for (T& x : Vec) { is >> x; }
	return is;
}
template<typename V, typename H>void resize(vector<V>& vec, const H head) {
	vec.resize(head);
}
template<typename V, typename H, typename ... T>void resize(vector<V>& vec, const H& head, const T ... tail) {
	vec.resize(head);
	for (auto& v : vec) { resize(v, tail...); }
}
template<ll mod> struct ModInt {
	//けんちょんさん作modをとる整数
	//名前だけ変えた
	//これがないと二項係数が動かないので注意
	//普通に計算するとmodがとれる
	long long val;
	constexpr ModInt(long long v = 0) noexcept : val(v % mod) {
		if (val < 0) v += mod;
	}
	constexpr int getmod() { return mod; }
	constexpr ModInt operator - () const noexcept {
		return val ? mod - val : 0;
	}
	constexpr ModInt operator + (const ModInt& r) const noexcept { return ModInt(*this) += r; }
	constexpr ModInt operator - (const ModInt& r) const noexcept { return ModInt(*this) -= r; }
	constexpr ModInt operator * (const ModInt& r) const noexcept { return ModInt(*this) *= r; }
	constexpr ModInt operator / (const ModInt& r) const noexcept { return ModInt(*this) /= r; }
	constexpr ModInt& operator += (const ModInt& r) noexcept {
		val += r.val;
		if (val >= mod) val -= mod;
		return *this;
	}
	constexpr ModInt& operator -= (const ModInt& r) noexcept {
		val -= r.val;
		if (val < 0) val += mod;
		return *this;
	}
	constexpr ModInt& operator *= (const ModInt& r) noexcept {
		val = val * r.val % mod;
		return *this;
	}
	constexpr ModInt& operator /= (const ModInt& r) noexcept {
		long long a = r.val, b = mod, u = 1, v = 0;
		while (b) {
			long long t = a / b;
			a -= t * b; swap(a, b);
			u -= t * v; swap(u, v);
		}
		val = val * u % mod;
		if (val < 0) val += mod;
		return *this;
	}
	constexpr bool operator == (const ModInt& r) const noexcept {
		return this->val == r.val;
	}
	constexpr bool operator != (const ModInt& r) const noexcept {
		return this->val != r.val;
	}
	friend ostream& operator << (ostream &os, const ModInt<mod>& x) noexcept {
		return os << x.val;
	}
	friend istream& operator >> (istream &is, ModInt<mod>& x) noexcept {
		return is >> x.val;
	}
	friend constexpr ModInt<mod> modpow(const ModInt<mod> &a, long long n) noexcept {
		if (n == 0) return 1;
		auto t = modpow(a, n / 2);
		t = t * t;
		if (n & 1) t = t * a;
		return t;
	}
};
template<class T> struct nCk {
	//けんちょんさん作二項係数ライブラリ
	//名前だけ変えた
	//nCk<ModInt<MOD>> c(200000);
	//↑のような感じで初期化
	vector<T> fact_, inv_, finv_;
	constexpr nCk() {}
	constexpr nCk(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
		init(n);
	}
	constexpr void init(int n) noexcept {
		fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
		int MOD = fact_[0].getmod();
		for (int i = 2; i < n; i++) {
			fact_[i] = fact_[i - 1] * i;
			inv_[i] = -inv_[MOD%i] * (MOD / i);
			finv_[i] = finv_[i - 1] * inv_[i];
		}
	}
	constexpr T com(int n, int k) const noexcept {
		if (n < k || n < 0 || k < 0) return 0;
		return fact_[n] * finv_[k] * finv_[n - k];
	}
	constexpr T fact(int n) const noexcept {
		if (n < 0) return 0;
		return fact_[n];
	}
	constexpr T inv(int n) const noexcept {
		if (n < 0) return 0;
		return inv_[n];
	}
	constexpr T finv(int n) const noexcept {
		if (n < 0) return 0;
		return finv_[n];
	}
};

int dx[] = { 0,1,-1, 0,1,-1, 1,-1 };    //i<4:4way i<8:8way
int dy[] = { 1,0, 0,-1,1,-1,-1, 1 };

ll PowMod(ll n, ll k, ll mod) {
	//繰り返し2乗法
	//n^kをmodで求める
	ll r = 1;

	for (; k > 0; k >>= 1) {
		if (k & 1) {
			r = (r * n) % mod;
		}
		n = (n * n) % mod;
	}
	return r;
}
ll Gcd(ll a, ll b) {//最大公約数
	return b != 0 ? Gcd(b, a % b) : a;
}
ll Lcm(ll a, ll b) {//最小公倍数
	return a / Gcd(a, b) * b;
}
vector<string> Split(string s, string t) {
	//文字列を文字列で分割する
	vector<string> v;
	int p = s.find(t);
	if (p != s.npos) {
		v.push_back(s.substr(0, p));
		s = s.substr(p + t.size());
	}
	v.push_back(s);
	return v;
}
vector<int> Lis(const vector<int>& a) {
	//#define Index_of(as, x) distance(as.begin(), lower_bound(as.begin(), as.end(), x))
#define Index_of(as, x) distance(as.begin(), upper_bound(as.begin(), as.end(), x))
//upper_boundを使用すると、重複を許した最長増加部分列になる
//-1倍した値を入れれば、最長減少部分列になる
	const int n = a.size();
	vector<int> A(n, INF);
	vector<int> id(n);
	for (int i = 0; i < n; ++i) {
		id[i] = Index_of(A, a[i]);
		A[id[i]] = a[i];
	}
	int m = *max_element(id.begin(), id.end());
	vector<int> b(m + 1);
	for (int i = n - 1; i >= 0; --i)
		if (id[i] == m) b[m--] = a[i];
	return b;//最長部分列のどれか1つ
}
string ReplaceString(string s, string target, string replacestring) {
	//1: 置き換え対象
	//2: 検索対象
	//3: 置き換える内容
	string::size_type  Pos(s.find(target));

	while (Pos != string::npos) {
		s.replace(Pos, target.length(), replacestring);
		Pos = s.find(target, Pos + replacestring.length());
	}

	return s;
}
string LcsAlphabeticalMinOrder(string a, string b) {
	if (a.size() < b.size()) { swap(a, b); }

	int n = a.size(), m = b.size();

	vector<string> dp(m + 1);

	for (int i = 0; i < n; i++) {
		vector<string> to(m + 1);
		for (int j = 0; j < m; j++) {
			if (a[i] == b[j]) {
				to[j + 1] = dp[j] + a[i];
			}
			else {
				if (to[j].size() > dp[j + 1].size()) { to[j + 1] = to[j]; }
				else if (to[j].size() < dp[j + 1].size()) { to[j + 1] = dp[j + 1]; }
				else if (to[j] < dp[j + 1]) { to[j + 1] = to[j]; }
				else { to[j + 1] = dp[j + 1]; }
			}
		}
		dp.swap(to);
	}
	return dp[m];
}
string Lcs(const string &s, const string &t) {
	int dp[3001][3001];
	int n = s.size();
	int m = t.size();
	for (int i = 1; i <= n; i++) {
		for (int j = 1; j <= m; j++) {
			if (s[i - 1] == t[j - 1]) {
				dp[i][j] = dp[i - 1][j - 1] + 1;
			}
			else {
				dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]);
			}
		}
	}
	string ans = "";
	int i = s.size(), j = t.size();
	while (i > 0 && j > 0) {
		if (s[i - 1] == t[j - 1]) {
			ans += s[i - 1];
			i--;
			j--;
		}
		else if (dp[i - 1][j] >= dp[i][j - 1])
			i--;
		else
			j--;
	}
	reverse(ans.begin(), ans.end());
	return ans;
}
vector<int> LcsInteger(const vector<int> &a, const vector<int> &b) {
#define index_of(as, x) distance(as.begin(), lower_bound(as.begin(), as.end(), x))
	struct node {
		int value;
		node *next;
		node(int value, node *next) : value(value), next(next) { }
	};
	const int n = a.size(), m = b.size();
	map< int, vector<int> > M;
	for (int j = m - 1; j >= 0; --j)
		M[b[j]].push_back(j);
	vector<int> xs(n + 1, INF); xs[0] = -INF;
	vector< node* > link(n + 1);
	for (int i = 0; i < n; ++i) {
		if (M.count(a[i])) {
			vector<int> ys = M[a[i]];
			for (int j = 0; j < (int)ys.size(); ++j) {
				int k = index_of(xs, ys[j]);
				xs[k] = ys[j];
				link[k] = new node(b[ys[j]], link[k - 1]);
			}
		}
	}
	vector<int> c;
	int l = index_of(xs, INF - 1) - 1;
	for (node *p = link[l]; p; p = p->next)
		c.push_back(p->value);
	reverse(c.begin(), c.end());
	return c;
}
bool IsPrime(ll n) {
	//素数かどうかを判定
	//true 素数
	if (n < 2)return false;
	for (ll i = 2; i*i <= n; i++)if (!(n%i))return false;
	return true;
}
vector<bool> Eratosthenes(int n) {
	//エラトステネスの篩
	//返り値はbool
	//return res;として、返り値の型をvector<int>とすれば、
	//素数だけを取り出せる
	vector<int> res;
	vector<bool> Prime(n + 1, true);
	Prime[0] = Prime[1] = false;
	for (int i = 2; i*i <= n; i++) {
		if (Prime[i]) {
			for (int j = 2; i*j <= n; j++) {
				Prime[i*j] = false;
			}
		}
	}
	for (int i = 2; i <= n; i++) {
		if (Prime[i]) {
			res.emplace_back(i);
		}
	}
	return Prime;
}
ll MergeCount(vector<int> &a) {
	//反転数を数える
	ll count = 0;
	int n = a.size();
	if (n > 1) {
		vector<int> b(a.begin(), a.begin() + n / 2);
		vector<int> c(a.begin() + n / 2, a.end());
		count += MergeCount(b);
		count += MergeCount(c);
		for (int i = 0, j = 0, k = 0; i < n; ++i)
			if (k == c.size())       a[i] = b[j++];
			else if (j == b.size())  a[i] = c[k++];
			else if (b[j] <= c[k])   a[i] = b[j++];
			else { a[i] = c[k++]; count += n / 2 - j; }
	}
	return count;
}
bool  WarshallFloyd(vector<vector<ll>> &c) {
	//ワーシャルフロイド法
	//全ての頂点間の最短距離を求める
	//falseの時、負の閉路検出
	int V = c.size();
	for (int i = 0; i < V; i++) {
		c[i][i] = 0;
	}

	for (int i = 0; i < V; i++) {
		for (int j = 0; j < V; j++) {
			for (int k = 0; k < V; k++) {
				if (c[j][k] > c[j][i] + c[i][k]) {
					c[j][k] = c[j][i] + c[i][k];
				}
			}
		}
	}

	for (int i = 0; i < V; i++) {
		if (c[i][i] < 0) {
			return false;
		}
	}



	return true;
}
vector<ll> Dijkstra(int i, vector<vector<edge>> graph) {
	//i:始点
	//graph:重み付きグラフ
	int n = graph.size();
	vector<ll> d(n, LONGINF);
	d[i] = 0;
	priority_queue<pair<ll, int>, vector<pair<ll, int>>, greater<pair<ll, int>>> q;
	q.push(make_pair(0, i));//第一引数:コスト 第二引数:頂点
	while (!q.empty()) {
		pair<ll, int> p = q.top();
		q.pop();
		int v = p.second;
		if (d[v] < p.first) {
			continue;
		}
		for (auto x : graph[v]) {
			if (d[x.to] > d[v] + x.cost) {
				d[x.to] = d[v] + x.cost;
				q.push(make_pair(d[x.to], x.to));
			}
		}
	}
	return d;
}
bool BellmanFord(int start, int V, int E, vector<edge> Edge, vector<ll> &d) {
	//第一引数:start 始点
	//第二引数:V 頂点数
	//第三引数:E 辺の数
	//第四引数:Edge 辺の重み付きのグラフ
	//第五引数:d 各頂点への距離を入れる配列(答えが入る)
	//負の閉路検出でfalseが返り値
	resize(d, V);
	fill(d.begin(), d.end(), LONGINF);
	d[start] = 0;
	vector<bool> t(V, false);

	for (int i = 0; i < V - 1; i++) {
		for (int j = 0; j < E; j++) {
			edge e = Edge[j];
			if (d[e.from] == LONGINF) { continue; }
			if (d[e.to] > d[e.from] + e.cost) {
				d[e.to] = d[e.from] + e.cost;
			}
		}
	}

	for (int i = 0; i < V; i++) {
		for (int j = 0; j < E; j++) {
			edge e = Edge[j];
			if (d[e.from] == LONGINF) { continue; }
			if (d[e.to] > d[e.from] + e.cost) {
				d[e.to] = d[e.from] + e.cost;
				t[e.to] = true;
				/*
				if (i == V - 1) {//どこかに閉路があることを感知する
					return false;
				}
				*/
			}
			if (t[e.from]) {
				t[e.to] = true;
			}
		}
	}

	if (t[V - 1]) {
		//V-1は頂点番号n-1で、始点からn-1までに負の閉路を検出したい場合には、
		//コメントアウトを解除すること。
		return false;
	}

	return true;
}
bool TopologicalSort(const vector<vector<edge>> &g, vector<int> &ans) {
	//トポロジカルソート
	//trueが帰る時、トポロジカルソートが成功し、その結果がansに渡される
	//falseはトポロジカルソートの失敗
	int n = g.size(), k = 0;
	vector<int> ord(n), in(n);
	for (auto &es : g) {
		for (auto &e : es) {
			in[e.to]++;
		}
	}
	queue<int> q;
	for (int i = 0; i < n; ++i) {
		if (in[i] == 0) q.push(i);
	}
	while (!q.empty()) {
		int v = q.front();
		q.pop();
		ord[k++] = v;
		for (auto &e : g[v]) {
			if (--in[e.to] == 0) q.push(e.to);
		}
	}
	ans = ord;
	if (*max_element(in.begin(), in.end()) == 0) { return true; }
	return false;
}
vector<int> ArticulationNode(const vector<vector<edge>>& g) {
	//グラフの関節点を列挙する
	//最後の2行で、erace uniqueをしない場合は、その分割によって何個のグラフに分かれるかを判定できる(要チェック)。
	int n = g.size(), idx;
	vector<int> low(n), ord(n), art;
	function<void(int)> DFS = [&](int v) {
		low[v] = ord[v] = ++idx;
		for (auto& e : g[v]) {
			int w = e.to;
			if (ord[w] == 0) {
				DFS(w);
				low[v] = min(low[v], low[w]);
				if ((ord[v] == 1 && ord[w] != 2) || (ord[v] != 1 && low[w] >= ord[v])) {
					art.push_back(v);
				}
			}
			else {
				low[v] = min(low[v], ord[w]);
			}
		}
	};
	for (int u = 0; u < n; u++) {
		if (ord[u] == 0) {
			idx = 0;
			DFS(u);
		}
	}

	sort(art.begin(), art.end());//与えられた関節点をソート
	art.erase(unique(art.begin(), art.end()), art.end());//同じ関節点が複数存在することがある,

	return art;
}
vector<vector<edge>> ToRootTree(const vector<vector<edge>> &g, int r) {
	//根付き木へ変換する。
	//動作未確認。
	int n = g.size();
	vector<vector<edge>> G(n);
	vector<int> ord(n, -1);

	queue<int> q;

	q.push(r);
	int k = 0;

	while (q.size()) {
		int u = q.front(); q.pop();

		for (auto &e : g[u]) {
			int v = e.to;
			if (ord[v] == -1) {
				ord[v] = k; k++;
				q.push(v);
				G[u].emplace_back(e);
			}
		}
	}

	return G;
}
edge TreeDiameter(const vector<vector<edge>> &g) {
	//重み付きグラフ(木)を受け取り、その木の直径を求める
	//返り値はfrom,to,costを持った構造体

	int start = 0;//どの始点から始めても良いので、0から始める

	static const auto bfs = [](const vector<vector<edge>> &g, int s, queue<int> &q, vector<ll> &dist) {
		while (!q.empty()) { q.pop(); }
		q.push(s);
		int n = g.size();
		dist.assign(n, LONGINF);
		dist[s] = 0;
		while (q.size()) {
			int u = q.front();
			q.pop();
			for (auto &e : g[u]) {
				int v = e.to;
				if (dist[v] == LONGINF) {
					dist[v] = dist[u] + e.cost;
					q.push(v);
				}
			}
		}
		return dist;
	};
	vector<ll> dist;
	queue<int> q;
	bfs(g, start, q, dist);
	int n = g.size(), u = -1, v = -1;
	for (int i = 0; i < n; i++)
		if (dist[i] != LONGINF && (u == -1 || dist[i] > dist[u])) u = i;
	bfs(g, u, q, dist);
	for (int i = 0; i < n; i++)
		if (dist[i] != LONGINF && (v == -1 || dist[i] > dist[v])) v = i;
	ll d = dist[v];
	if (u > v) swap(u, v);//念のため辞書順
	return edge(u, v, d);
}
void add_edge(vector<vector<edge>> &g, int a, int b, ll cost, ll cap) {
	//graph(vector<vector<edge>>)に対して無向辺として辺を貼る関数
	//graph以外には当然使えないので注意
	g[a].emplace_back(a, b, cost, cap);
	g[b].emplace_back(b, a, cost, cap);
}
pair<vector<int>, vector<edge>> bridge(const vector<vector<edge>>& g) {
	//グラフの橋となる辺を列挙する
	//戻り値のfirst:二重辺連結成分分解の番号となる
	//戻り値のsecond:橋となる辺を列挙したもの
	const int n = g.size();
	int idx = 0, s = 0, t = 0, k = 0;
	vector<int> ord(n, -1), onS(n), stk(n), roots(n), cmp(n);
	vector<edge> brdg;
	function<void(int, int)> dfs = [&](int v, int u) {
		ord[v] = idx++;
		stk[s++] = v;
		onS[v] = true;
		roots[t++] = v;
		for (auto& e : g[v]) {
			int w = e.to;
			if (ord[w] == -1) { dfs(w, v); }
			else if (u != w && onS[w]) {
				while (ord[roots[t - 1]] > ord[w]) { --t; }
			}
		}
		if (v == roots[t - 1]) {
			brdg.emplace_back(u, v, 0);
			while (1) {
				int w = stk[--s];
				onS[w] = false;
				cmp[w] = k;
				if (v == w) break;
			}
			--t;
			++k;
		}
	};
	for (int u = 0; u < n; u++) {
		if (ord[u] == -1) {
			dfs(u, n);
			brdg.pop_back();
		}
	}
	return make_pair(cmp, brdg);
}

class UnionFind {
	//satanicさん作 UnionFind
	//追加機能:forest_num
	//forestは、全体に含まれる木の数を表す
private:
	std::vector<int> parent;
	std::vector<int> height;
	std::vector<int> m_size;
	int forest_num;
public:
	UnionFind(int size_) : parent(size_), height(size_, 0), m_size(size_, 1) {
		forest_num = size_;
		for (int i = 0; i < size_; ++i) parent[i] = i;
	}
	void init(int size_) {
		parent.resize(size_);
		height.resize(size_, 0);
		m_size.resize(size_, 1);
		forest_num = size_;
		for (int i = 0; i < size_; ++i) parent[i] = i;
	}
	int find(int x) {
		if (parent[x] == x) return x;
		return parent[x] = find(parent[x]);
	}
	void unite(int x, int y) {
		x = find(x);
		y = find(y);
		if (x == y) return;
		int t = size(x) + size(y);
		m_size[x] = m_size[y] = t;
		if (height[x] < height[y]) parent[x] = y;
		else parent[y] = x;
		if (height[x] == height[y]) ++height[x];
		forest_num--;
	}
	bool same(int x, int y) {
		return find(x) == find(y);
	}
	int size(int x) {
		if (parent[x] == x) return m_size[x];
		return size(parent[x] = find(parent[x]));
	}
	int forest() {
		return forest_num;
	}
};
class Dinic {
	//最大流を求める
#define FLOW int
private:
	int n, s, t;
	vector<int> level, prog, que;
	vector<vector<FLOW>> cap, flow;
	vector<vector<int>> g;
	FLOW inf;
public:
	Dinic(const vector<vector<edge>> &graph) :
		n(graph.size()),
		cap(n, vector<FLOW>(n)),
		flow(n, vector<FLOW>(n)),
		g(n, vector<int>()),
		inf(INF) {
		for (int i = 0; i < n; i++) {
			for (auto &e : graph[i]) {
				int u = e.from, v = e.to;
				ll c = e.capa;
				cap[u][v] += c;
				cap[v][u] += c;
				flow[v][u] += c;
				g[u].push_back(v);
				g[v].push_back(u);
			}
		}
	}
	inline FLOW residue(int u, int v) { return cap[u][v] - flow[u][v]; }
	FLOW solve(int s_, int t_) {
		this->t = t_, this->s = s_;
		que.resize(n + 1);
		FLOW res = 0;
		while (levelize()) {
			prog.assign(n, 0);
			res += augment(s, inf);
		}
		return res;
	}
	bool levelize() {
		int l = 0, r = 0;
		level.assign(n, -1);
		level[s] = 0;
		que[r++] = s;
		while (l != r) {
			int v = que[l++];
			if (v == t) break;
			for (const int &d : g[v])
				if (level[d] == -1 && residue(v, d) != 0) {
					level[d] = level[v] + 1;
					que[r++] = d;
				}
		}
		return level[t] != -1;
	}
	FLOW augment(int v, FLOW lim) {
		FLOW res = 0;
		if (v == t) return lim;
		for (int &i = prog[v]; i < (int)g[v].size(); i++) {
			const int &d = g[v][i];
			if (residue(v, d) == 0 || level[v] >= level[d]) continue;
			const FLOW aug = augment(d, min(lim, residue(v, d)));
			flow[v][d] += aug;
			flow[d][v] -= aug;
			res += aug;
			lim -= aug;
			if (lim == 0) break;
		}
		return res;
	}
};
class MinimumCostFlow {
private:

	using Flow = ll;
	using Cost = ll;
	struct Edge {
		int d;
		Flow c, f;
		Cost w;
		int r, is_r;
		Edge(int d_, Flow c_, Flow f_, Cost w_, int r_, bool is_r_)
			: d(d_), c(c_), f(f_), w(w_), r(r_), is_r(is_r_) {}
	};
	int n;
	vector<vector<Edge>> g;

public:

	MinimumCostFlow(int n_) : n(n_), g(vector<vector<Edge>>(n_)) {}

	void add_edge(int src, int dst, Flow cap, Cost cost) {  // 有向辺
		int rsrc = g[dst].size();
		int rdst = g[src].size();
		g[src].emplace_back(dst, cap, 0, cost, rsrc, false);
		g[dst].emplace_back(src, cap, cap, -cost, rdst, true);
	}

	Cost solve(int s, int t, Flow f) {
		Cost res = 0;

		vector<Cost> h(n + 10), dist(n);
		vector<int> prevv(n + 10), preve(n + 10);

		using pcv = pair<Cost, int>;
		priority_queue<pcv, vector<pcv>, greater<pcv> > q;
		fill(h.begin(), h.end(), 0);
		while (f > 0) {
			fill(dist.begin(), dist.end(), LONGINF);
			dist[s] = 0;
			q.emplace(0, s);
			while (q.size()) {
				Cost cd;
				int v;
				tie(cd, v) = q.top();
				q.pop();
				if (dist[v] < cd) continue;
				for (int i = 0; i < (int)(g[v].size()); ++i) {
					Edge &e = g[v][i];
					if (residue(e) == 0) continue;
					if (dist[e.d] + h[e.d] > cd + h[v] + e.w) {
						dist[e.d] = dist[v] + e.w + h[v] - h[e.d];
						prevv[e.d] = v;
						preve[e.d] = i;
						q.emplace(dist[e.d], e.d);
					}
				}
			}

			if (dist[t] == LONGINF) return -1;  // 経路が見つからなかった

			// s-t 間を最短路に沿って目一杯流す
			for (int i = 0; i < n; ++i) h[i] += dist[i];
			Flow d = f;
			for (int v = t; v != s; v = prevv[v]) {
				chmin(d, residue(g[prevv[v]][preve[v]]));
			}
			f -= d;
			res += d * h[t];
			for (int v = t; v != s; v = prevv[v]) {
				Edge &e = g[prevv[v]][preve[v]];
				e.f += d;
				g[v][e.r].f -= d;
			}
		}
		return res;
	}

	Flow residue(const Edge &e) { return e.c - e.f; }

	// 流量を表示
	void show() {
		for (int i = 0; i < n; ++i) {
			for (int j = 0; j < (int)(g[i].size()); ++j) {
				Edge &e = g[i][j];
				if (e.is_r) continue;
				cout << i << "->" << e.d << "(flow:" << e.f << ")" << endl;
			}
		}
	}
};
class BipartiteMatching {
private:
	int V;
	vector<int> match;
	vector<bool> used;
	vector<vector<int>> g;
	vector<pair<int, int>> match_pair;

	bool dfs(int v) {
		used[v] = true;
		for (int i = 0; i < (int)g[v].size(); i++) {
			int u = g[v][i];
			int w = match[u];
			if (w < 0 || !used[w] && dfs(w)) {
				match[v] = u;
				match[u] = v;
				match_pair.emplace_back(make_pair(u, v));
				return true;
			}
		}
		return false;
	}

public:
	BipartiteMatching(int n) {
		V = n;
		resize(match, n);
		resize(used, n);
		resize(g, n);
	}

	void add_edge(int u, int v) {
		g[u].emplace_back(v);
		g[v].emplace_back(u);
	}

	int MatchingSolve() {
		int res = 0;
		fill(match.begin(), match.end(), -1);

		for (int v = 0; v < V; v++) {
			if (match[v] < 0) {
				fill(used.begin(), used.end(), false);
				if (dfs(v)) {
					res++;
				}
			}
		}
		return res;
	}

	vector<pair<int, int>> get_pair() {
		for (auto x : match_pair) {
			cout << x.first << "  " << x.second << endl;
		}
		return match_pair;
	}

};
class Lca {
private:
	int n;
	int log2_n;
	vector<vector<int>> parent;
	vector<int> depth;

	void dfs(const vector<vector<edge>> &g, int v, int p, int d) {
		parent[0][v] = p;
		depth[v] = d;
		for (auto &e : g[v]) {
			if (e.to != p) { dfs(g, e.to, v, d + 1); }
		}
	}

public:

	Lca(const vector<vector<edge>> &g, int root) {
		n = g.size();
		log2_n = (int)log2(n) + 1;
		resize(parent, log2_n, n);
		resize(depth, n);

		dfs(g, root, -1, 0);

		for (int k = 0; k + 1 < log2_n; k++) {
			for (int v = 0; v < (int)g.size(); v++) {
				if (parent[k][v] < 0) {
					parent[k + 1][v] = -1;
				}
				else {
					parent[k + 1][v] = parent[k][parent[k][v]];
				}
			}
		}

	}

	int get_lca(int u, int v) {
		if (depth[u] > depth[v]) { swap(u, v); }//u≦v

		for (int k = 0; k < log2_n; k++) {
			if ((depth[v] - depth[u]) >> k & 1) {
				v = parent[k][v];
			}
		}
		if (u == v) { return u; }

		for (int k = log2_n - 1; k >= 0; k--) {
			if (parent[k][u] != parent[k][v]) {
				u = parent[k][u];
				v = parent[k][v];
			}
		}
		return parent[0][u];
	}

	int get_depth(int v) {
		return depth[v];
	}
};
class DAG {
	//有向グラフをゴニョゴニョするクラス
	//機能:longest_path 有向パスの最長経路(含まれる辺の数)を求める
private:
	int n;
	vector<vector<edge>> g;
	vector<int> visited;
	vector<int> dp;
	vector<int> topological;

	int dfs(int s) {
		if ((int)g[s].size() == 0) {
			return 1;
		}
		if (dp[s] > 0) { return dp[s]; }

		int mx = 1;
		for (auto j : g[s]) {
			if (visited[j.to] == 0) {
				visited[j.to] = 1;
				int l = 0;
				l = dfs(j.to);
				chmax(mx, l);
			}
			else {
				chmax(mx, dp[j.to]);
			}
		}
		return dp[s] = mx + 1;
	}
public:
	DAG(const vector<vector<edge>> &f) {
		g = f;
		n = f.size();
		resize(visited, n + 1);
		fill(visited.begin(), visited.end(), 0);
		resize(dp, n + 1);
		fill(dp.begin(), dp.end(), -1);
		resize(topological, n);
	}
	DAG(int x) {
		n = x;
		resize(g, n);
		resize(visited, n + 1);
		fill(visited.begin(), visited.end(), 0);
		resize(dp, n + 1);
		fill(dp.begin(), dp.end(), -1);
	}

	void add_edge(int a, int b) {
		g[a].emplace_back(a, b);
	}
	void add_edge(int a, int b, ll c) {
		g[a].emplace_back(a, b, c);
	}
	void add_edge(int a, int b, ll c, ll d) {
		g[a].emplace_back(a, b, c, d);
	}

	int longest_path() {
		int mx = -1;
		for (int i = 0; i < n; i++) {
			int h = 0;
			if (visited[i] == 0) {
				h = dfs(i);
				chmax(mx, h);
			}
		}
		return mx - 1;
	}

	bool TopologicalSort() {
		//trueが帰る時、トポロジカルソートが成功
		//falseはトポロジカルソートの失敗
		int k = 0;
		vector<int> ord(n), in(n);
		for (auto &es : g) {
			for (auto &e : es) {
				in[e.to]++;
			}
		}
		queue<int> q;
		for (int i = 0; i < n; ++i) {
			if (in[i] == 0) q.push(i);
		}
		while (!q.empty()) {
			int v = q.front();
			q.pop();
			ord[k++] = v;
			for (auto &e : g[v]) {
				if (--in[e.to] == 0) { q.push(e.to); }
			}
		}
		topological = ord;
		if (*max_element(in.begin(), in.end()) == 0) { return true; }
		return false;
	}

};
class RangeMinimumUpdateQuerySegmentTree {
	//区間最小値を求めることが出来るセグメント木
	//query:指定した半開区間においての最小値を求める
	//update:指定した半開区間の値を変更する
	//RangeUpdateQueryも兼ねることが出来る(1つの要素の最小値はその要素のみ)
private:
	int n;
	ll inf = (1LL << 31) - 1;//2^31-1
	vector<ll> dat, lazy;

	void eval(int len, int k) {
		if (lazy[k] == inf) return;
		if (k * 2 + 1 < n * 2 - 1) {
			lazy[2 * k + 1] = lazy[k];
			lazy[2 * k + 2] = lazy[k];
		}
		dat[k] = lazy[k];
		lazy[k] = inf;
	}
public:
	RangeMinimumUpdateQuerySegmentTree() {}
	RangeMinimumUpdateQuerySegmentTree(int n_) {
		n = 1; while (n < n_) n *= 2;
		dat.assign(n * 2, inf);
		lazy.assign(n * 2, inf);
	}

	// [a, b)
	ll update(int a, int b, ll x, int k, int l, int r) {
		eval(r - l, k);
		if (b <= l || r <= a) return dat[k];
		if (a <= l && r <= b) {
			lazy[k] = x;
			return lazy[k];
		}
		return dat[k] = min(update(a, b, x, 2 * k + 1, l, (l + r) / 2), update(a, b, x, 2 * k + 2, (l + r) / 2, r));
	}
	ll update(int a, int b, ll x) { return update(a, b, x, 0, 0, n); }

	// [a, b)
	ll query(int a, int b, int k, int l, int r) {
		eval(r - l, k);
		if (b <= l || r <= a) return inf;
		if (a <= l && r <= b) return dat[k];
		ll vl = query(a, b, 2 * k + 1, l, (l + r) / 2);
		ll vr = query(a, b, 2 * k + 2, (l + r) / 2, r);
		return min(vl, vr);
	}
	ll query(int a, int b) { return query(a, b, 0, 0, n); }
};
class RangeSumQuerySegmentTree {
	//区間和を求めることが出来るセグメント木
	//update:指定した要素に加算
	//query:指定した半開区間の合計を返す
	//0-indexed
	//半開区間
private:
	struct Node {
		Node *left, *right;
		ll v;

		Node() : left(nullptr), right(nullptr), v(0) {}
	};
	Node *root;
	ll n, n0;
	ll query(ll a, ll b, Node *n, ll l, ll r) {
		if (a <= l && r <= b) {
			return n->v;
		}
		if (r <= a || b <= l) {
			return 0;
		}

		ll lv = n->left ? query(a, b, n->left, l, (l + r) >> 1) : 0;
		ll rv = n->right ? query(a, b, n->right, (l + r) >> 1, r) : 0;
		return (lv + rv) % MOD;
	}
public:
	RangeSumQuerySegmentTree(ll n) : n(n) {
		n0 = 1;
		while (n0 < n) n0 <<= 1;
		root = new Node();
	}
	~RangeSumQuerySegmentTree() {
		delete root; root = nullptr;
	}

	void update(ll k, ll x) {
		Node *n = root;
		ll l = 0, r = n0;
		n->v = (n->v + x) % MOD;
		while (r - l > 1) {
			ll m = (l + r) >> 1;
			if (k < m) {
				if (!n->left) n->left = new Node();
				n = n->left;

				r = m;
			}
			else {
				if (!n->right) n->right = new Node();
				n = n->right;

				l = m;
			}
			n->v = (n->v + x) % MOD;
		}
	}

	ll query(ll a, ll b) {
		return query(a, b, root, 0, n0);
	}

	ll lquery(ll b) {
		return query(0, b, root, 0, n0);
	}

	ll rquery(ll a) {
		return query(a, n0, root, 0, n0);
	}
};
class KDimensionalTree {
	//2Dの点がある区間内にあるかどうかを判定し、その点を列挙する
	//makeKDTreeで前計算
	//findが実際に探す部分
public:
	struct Node {
		int location;
		int p, l, r;
		Node() {}
	};
	struct Point {
		int id, x, y;
		Point() {}
		Point(int i, int a, int b) {
			id = i;
			x = a;
			y = b;
		}
		bool operator<(const Point &p)const {
			return id < p.id;
		}
		void print() {
			cout << id << endl;
		}
	};
	static const ll NIL = -1;
	static bool lessX(const Point &p1, const Point &p2) { return p1.x < p2.x; }
	static bool lessY(const Point &p1, const Point &p2) { return p1.y < p2.y; }

	int N;
	vector<Point> P;
	vector<Node> T;
	int np;

	KDimensionalTree() {}
	KDimensionalTree(int N) { init(N); }

	void init(int n) {
		N = n;
		P.clear();
		T.clear();
		resize(P, N);
		resize(T, N);
		np = 0;
	}

	int makeKDTree(int l, int r, int depth) {
		if (l >= r) { return NIL; }
		int mid = (l + r) / 2;
		int t = np++;
		if (depth & 1) {
			sort(P.begin() + l, P.begin() + r, lessY);
		}
		else {
			sort(P.begin() + l, P.begin() + r, lessX);
		}
		T[t].location = mid;
		T[t].l = makeKDTree(l, mid, depth + 1);
		T[t].r = makeKDTree(mid + 1, r, depth + 1);
		return t;
	}
	void find(int v, int sx, int tx, int sy, int ty, int depth, vector<Point> &ans) {
		int x = P[T[v].location].x;
		int y = P[T[v].location].y;
		if (sx <= x && x <= tx && sy <= y && y <= ty) {
			ans.push_back(P[T[v].location]);
		}
		if (depth % 2 == 0) {
			if (T[v].l != NIL) {
				if (sx <= x) find(T[v].l, sx, tx, sy, ty, depth + 1, ans);
			}
			if (T[v].r != NIL) {
				if (x <= tx) find(T[v].r, sx, tx, sy, ty, depth + 1, ans);
			}
		}
		else {
			if (T[v].l != NIL) {
				if (sy <= y) find(T[v].l, sx, tx, sy, ty, depth + 1, ans);
			}
			if (T[v].r != NIL) {
				if (y <= ty) find(T[v].r, sx, tx, sy, ty, depth + 1, ans);
			}
		}
	}
	void add_point(int i, int x, int y) {
		P[i].id = i;
		P[i].x = x;
		P[i].y = y;
		T[i].l = T[i].r = T[i].p = NIL;
	}
};
class RangeAddQuerySegmentTree {
	//区間に足し算をする機能と、ある要素を参照する機能があるセグメント木
private:
	int n;
	vector<ll> data;
public:
	RangeAddQuerySegmentTree() {}
	RangeAddQuerySegmentTree(int N) {
		n = N;
		resize(data, n + 1);
		fill(data.begin(), data.end(), 0);
	}

	void add(int i, ll x) {
		while (i) {
			data[i] += x;
			i -= (i&-i);
		}
	}
	void add(int i, int j, ll x) {
		add(j, x);
		add(i - 1, -x);
	}

	ll get(int i) {
		ll res = 0;
		while (i <= n) {
			res += data[i];
			i += (i&-i);
		}
		return res;
	}
};
class RangeSumAddQuerySegmentTree {
	//update:ある区間に対して足し算を行う機能
	//query:ある区間に対して合計値を求める機能
private:
	vector<ll> bit0, bit1;
	int n;

	ll sum(const vector<ll> &b, int i) {
		ll s = 0;
		while (i > 0) {
			s += b[i];
			i -= (i&-i);
		}
		return s;
	}

	void add(vector<ll> &b, int i, ll v) {
		while (i <= n) {
			b[i] += v;
			i += (i&-i);
		}
	}
public:
	RangeSumAddQuerySegmentTree() {}
	RangeSumAddQuerySegmentTree(int N) {
		n = N;
		resize(bit0, n + 1);
		resize(bit1, n + 1);
		fill(bit0.begin(), bit0.end(), 0);
		fill(bit1.begin(), bit1.end(), 0);
	}

	void update(int l, int r, ll x) {
		add(bit0, l, -x * (l - 1));
		add(bit1, l, x);
		add(bit0, r + 1, x*r);
		add(bit1, r + 1, -x);
	}

	ll query(int l, int r) {
		ll res = 0;
		res += sum(bit0, r) + sum(bit1, r)*r;
		res -= sum(bit0, l - 1) + sum(bit1, l - 1)*(l - 1);
		return res;
	}
};



int main() {

	int n, m; cin >> n >> m;
	priority_queue<ll> q;
	for (int i = 0; i < n; i++) {
		ll x; cin >> x;
		q.push(x);
	}

	for (int i = 0; i < m; i++) {
		ll x = q.top(); q.pop();
		q.push(x / 2);
	}
	ll ans = 0;
	while (q.size()) {
		ans += q.top(); q.pop();
	}

	cout << ans << endl;
	


	return 0;
}


Measured memory_bytes: 1400