Problem: You are given two strings s and t consisting of lowercase Latin letters. The length of t is 2 (i.e. this string consists only of two characters).

In one move, you can choose any character of s and replace it with any lowercase Latin letter. More formally, you choose some i and replace s_i (the character at the position i) with some character from 'a' to 'z'.

You want to do no more than k replacements in such a way that maximizes the number of occurrences of t in s as a subsequence.

Recall that a subsequence is a sequence that can be derived from the given sequence by deleting zero or more elements without changing the order of the remaining elements.

Input

The first line of the input contains two integers n and k (2 ≤ n ≤ 200; 0 ≤ k ≤ n) — the length of s and the maximum number of moves you can make. The second line of the input contains the string s consisting of n lowercase Latin letters. The third line of the input contains the string t consisting of two lowercase Latin letters.

Output

Print one integer — the maximum possible number of occurrences of t in s as a subsequence if you replace no more than k characters in s optimally.

Examples

Input


4 2
bbaa
ab


Output


3


Input


7 3
asddsaf
sd


Output


10


Input


15 6
qwertyhgfdsazxc
qa


Output


16


Input


7 2
abacaba
aa


Output


15

Note

In the first example, you can obtain the string "abab" replacing s_1 with 'a' and s_4 with 'b'. Then the answer is 3.

In the second example, you can obtain the string "ssddsdd" and get the answer 10.

In the fourth example, you can obtain the string "aaacaaa" and get the answer 15.

Solution (in JAVA):

Solution:
import java.io.*;
import java.math.BigInteger;
import java.util.*;


public class Main {
	public static void main(String[] args) throws IOException {
		InputStream inputStream = System.in;
		OutputStream outputStream = System.out;
		PrintWriter out = new PrintWriter(outputStream);
		InputReader in = new InputReader(inputStream);
		
//		for(int i=4;i<=4;i++) {
//			InputStream uinputStream = new FileInputStream("timeline.in");
			//String f = i+".in";
			//InputStream uinputStream = new FileInputStream(f);
//			InputReader in = new InputReader(uinputStream);
//			PrintWriter out = new PrintWriter(new BufferedWriter(new FileWriter("timeline.out")));
//		}
		Task t = new Task();
		t.solve(in, out);
		out.close();
	}	
	
	static class Task{

		public void solve(InputReader in, PrintWriter out) throws IOException {
//			String a[] = {"####F","#C...","M...."};
//			String b[] = {"M.C...F"};
//			String c[] = {"M.C...F"};
//			String d[] = {"C...#","...#F","....#","M...."};
//			String e[] = {".M...","..#..","#..#.","C#.#.","...#F"};
//			String f[] = {"#..C...","M....#.","######.","######.","######.","######F"};
//			//Dumper.print(canMouseWin(a,1,2));
//			//Dumper.print(canMouseWin(b,1,4));
//			//Dumper.print(canMouseWin(c,1,3));
//			//Dumper.print(canMouseWin(d,2,5));
//			//Dumper.print(canMouseWin(e,3,1));
//			Dumper.print(canMouseWin(f,1,5));
			
			int n = in.nextInt();
			int k = in.nextInt();
			char[] arr = in.next().toCharArray();
			String tmp = in.next();
			char a = tmp.charAt(0);
			char b = tmp.charAt(1);
			if(a==b) {
				int cnt = 0;
				for(char i:arr) if(i==a) cnt++;
				cnt = Math.min(n, cnt+k);
				out.println(cnt*(cnt-1)/2);
				return;
			}
			if(k==0) {
				int aa = 0;
				int ret = 0;
				for(char i:arr) {
					if(i==a) aa++;
					if(i==b) ret+=aa;
				}
				out.println(ret);
				return;
			}
			
			//long ti = System.nanoTime();
			//Dumper.print(System.nanoTime());
			
			int dp[][][] = new int[2][k+1][n+1];
			for(int i=0;i<2;i++) {
				for(int j=0;j<=k;j++) {
					for(int p=0;p<=n;p++) {
						dp[i][j][p] = -1;
					}
				}				
			}
			if(arr[0]==a) {
				dp[0][0][1] = 0;
				dp[0][1][0] = 0;
			}else if(arr[0]==b) {
				dp[0][0][0] = 0;
				dp[0][1][1] = 0;
			}else {
				dp[0][1][1] = 0;
				dp[0][1][0] = 0;
				dp[0][0][0] = 0;
			}
			
			
			for(int i=1;i<n;i++) {
				for(int j=0;j<=k;j++) {
					if(arr[i]==a) {
						for(int p=0;p<=n;p++) {
								dp[i%2][j][p]= Math.max(dp[i%2][j][p], dp[(i-1)%2][j][p]);
								if(p-1>=0) dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j][p-1]);
								if(j-1>=0&&dp[(i-1)%2][j-1][p]!=-1) dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j-1][p]+p);
						}					
					}else if(arr[i]==b) {
						for(int p=0;p<=n;p++) {
								dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j][p]);
								if(p-1>=0&&j-1>=0) dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j-1][p-1]);
								if(dp[(i-1)%2][j][p]!=-1) dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j][p]+p);
						}						
					}else {
						for(int p=0;p<=n;p++) {
								dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j][p]);
								if(p-1>=0&&j-1>=0) dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j-1][p-1]);
								if(j-1>=0&&dp[(i-1)%2][j-1][p]!=-1) dp[i%2][j][p] = Math.max(dp[i%2][j][p], dp[(i-1)%2][j-1][p]+p);
						}
					}
				}
			}
			
			int occ = 0;
			for(int i=0;i<=k;i++) {
				for(int p=0;p<=n;p++) {
					occ = Math.max(occ, dp[(n-1)%2][i][p]);
				}
			}
			//Dumper.print((System.nanoTime()-ti)/1000000);
			out.println(occ);
		}

		
	    public boolean canMouseWin(String[] grid, int catJump, int mouseJump) {
	        int n = grid.length;
	        int m = grid[0].length();
	        int cx = 0;
	        int cy = 0;
	        int mx = 0;
	        int my = 0;	  
	        int fx = 0;
	        int fy = 0;
	        char arr[][] = new char[n][m];
	        for(int i=0;i<n;i++) {
	        	arr[i] = grid[i].toCharArray();
	        	for(int j=0;j<m;j++) {
	        		if(arr[i][j]=='M') {
	        			mx = i;
	        			my = j;
	        		}
	        		if(arr[i][j]=='C') {
	        			cx = i;
	        			cy = j;
	        		}
	        		if(arr[i][j]=='F') {
	        			fx = i;
	        			fy = j;
	        		}	        		
	        	}
	        }
	        boolean can_jump[][][][] = new boolean[n][m][n][m];
	        for(int i=0;i<n;i++) {
	        	for(int j=0;j<m;j++) {
	        		for(int p=0;p<n;p++) {
	        			for(int q=0;q<m;q++) {
	        				if(i!=p&&j!=q) continue;
	        				if(i==p) {
	        					boolean flag = true;
	        					for(int k=Math.min(j, q);k<=Math.max(j, q);k++) {
	        						if(arr[i][k]=='#') {
	        							flag = false;
	        							break;
	        						}	        						
	        					}
	        					can_jump[i][j][p][q] = flag;
	        				}else {
	        					boolean flag = true;
	        					for(int k=Math.min(i, p);k<=Math.max(i, p);k++) {
	        						if(arr[k][j]=='#') {
	        							flag = false;	        						
	        							break;
	        						}
	        					}
	        					can_jump[i][j][p][q] = flag;	        					
	        				}
	        			}
	        		}
	        	}
	        }

	        int dir[][] = {{-1,0},{1,0},{0,-1},{0,1}};
	        boolean dp[][][][] = new boolean[n][m][n][m];
	        
	        dp[cx][cy][mx][my] = true;
	        dfs(0,can_jump,dp,arr,cx,cy,mx,my,fx,fy,catJump,mouseJump,dir);
	        for(int i=0;i<n;i++) {
	        	for(int j=0;j<m;j++) {
	        		if(dp[i][j][fx][fy]) return true;
	        	}
	        }
	        return false;
	    }
	    
	    void dfs(int turn, boolean can[][][][], boolean dp[][][][], char arr[][], int cx, int cy, int mx, int my, int fx, int fy, int cj, int mj, int dir[][]) {
	    	if(turn==0) {
	    		for(int i=0;i<4;i++) {
	    			for(int j=0;j<=mj;j++) {
	    				int nxt_mx = mx+dir[i][0]*j;
	    				int nxt_my = my+dir[i][1]*j;
	    				if(nxt_mx<0||nxt_my<0||nxt_mx>=arr.length||nxt_my>=arr[0].length) continue;
	    				if(arr[nxt_mx][nxt_my]=='#'||(nxt_mx==cx&&nxt_my==cy)||dp[cx][cy][nxt_mx][nxt_my]||!can[mx][my][nxt_mx][nxt_my]) continue;
	    				dp[cx][cy][nxt_mx][nxt_my] = true;
	    				dfs(1-turn,can,dp,arr,cx,cy,nxt_mx,nxt_my,fx,fy,cj,mj,dir);
	    			}
	    		}
	    	}else {
	    		if(cx==mx&&Math.abs(cy-my)<=cj) return;
	    		if(cy==my&&Math.abs(cx-mx)<=cj) return;
	    		for(int i=0;i<4;i++) {
	    			for(int j=0;j<=mj;j++) {
	    				int nxt_cx = cx+dir[i][0]*j;
	    				int nxt_cy = cy+dir[i][1]*j;
	    				if(nxt_cx<0||nxt_cy<0||nxt_cx>=arr.length||nxt_cy>=arr[0].length) continue;
	    				if(arr[nxt_cx][nxt_cy]=='#'||dp[nxt_cx][nxt_cy][mx][my]||!can[cx][cy][nxt_cx][nxt_cy]) continue;
	    				dp[nxt_cx][nxt_cy][mx][my] = true;
	    				dfs(1-turn,can,dp,arr,nxt_cx,nxt_cy,mx,my,fx,fy,cj,mj,dir);
	    			}
	    		}    		
	    	}
	    }
	    
	    class status {
	    	int cx,cy,mx,my;
	    	public status(int a, int b, int c, int d) {
	    		cx = a;
	    		cy = b;
	    		mx = c;
	    		my = d;
	    	}
	    }
	    
		

		 public Set<Integer> get_factor(int number) {
			 int n = number;
			 Set<Integer> primeFactors = new HashSet<Integer>();
			 for (int i = 2; i <= n/i; i++) { while (n % i == 0) { primeFactors.add(i); n /= i; } } if(n>1)
			 primeFactors.add(n);
			 return primeFactors;
		}

		private static int combx(int n, int k) {
			int comb[][] = new int[n+1][n+1];
			for(int i = 0; i <=n; i ++){
				comb[i][0] = comb[i][i] = 1;
				for(int j = 1; j < i; j++){
				comb[i][j] = comb[i-1][j] + comb[i-1][j-1];
				comb[i][j] %= 1000000007;
				}
			}
			return comb[n][k];
		}


		
		class trie_node{
			boolean end;
			int val;
			int lvl;
			trie_node zero;
			trie_node one;
			public trie_node() {
				zero = null;
				one = null;		
				end = false;
				val = -1;
				lvl = -1;
			}
		}
		class trie{
			trie_node root = new trie_node();			
			public void build(int x, int sz) {
				trie_node cur = root;				
				for(int i=sz;i>=0;i--) {
					int v = (x&(1<<i))==0?0:1;
					if(v==0&&cur.zero==null) {
						cur.zero = new trie_node();
					}
					if(v==1&&cur.one==null) {
						cur.one = new trie_node();
					}
					cur.lvl = i;
					if(i==0) {
						cur.end = true;
						cur.val = x;						
					}else {

						if(v==0)
							cur = cur.zero;
						else 
							cur = cur.one;
						cur.val = v;
						cur.lvl = i;						
					}
				}
			}
			int search(int num, int limit, trie_node r, int lvl) {
				if(r==null) return -1;
				if(r.end) return r.val;
				int f = -1;
				int num_val = (num&1<<lvl)==0?0:1;
				int limit_val = (limit&1<<lvl)==0?0:1;
				
				if(limit_val==1) {
					if(num_val==0) {
						int t = search(num,limit,r.one,lvl-1);
						if(t>f) return t;
						t = search(num,limit,r.zero,lvl-1);
						if(t>f) return t;						
					}else {
						int t = search(num,limit,r.zero,lvl-1);
						if(t>f) return t;
						t = search(num,limit,r.one,lvl-1);
						if(t>f) return t;						
					}					
				}else {
					int t = search(num,limit,r.zero,lvl-1);
					if(t>f) return t;
				}
				return f;
			}
		}
	    public int[] maximizeXor(int[] nums, int[][] queries) {
	    	int m = queries.length;
	        int ret[] = new int[m];
	        trie t = new trie();
	        int sz = 5;
	        for(int i:nums) t.build(i,sz);
	        int p = 0;	        
	        for(int x[]:queries) {
	        	if(p==1) {
	        		Dumper.print("here");
	        	}
	        	ret[p++] = t.search(x[0], x[1], t.root, sz);
	        }
	        return ret;
	    }		

		class edge implements Comparable<edge>{
			int id,f,t; int len;int short_len;
			public edge(int a, int b, int c, int d, int e) {
				f=a;t=b;len=c;id=d;short_len = e;
			}
			@Override
			public int compareTo(edge t) {				
				if(this.len-t.len>0) return 1;
				else if(this.len-t.len<0) return -1;
				return 0;
			}
		}
		
		static class lca_naive{
			int n;
			ArrayList<edge>[] g;
			int lvl[];
			int pare[];
			int dist[];
			public lca_naive(int t, ArrayList<edge>[] x) {
				n=t;
				g=x; 
				lvl = new int[n];
				pare = new int[n];
				dist = new int[n];
			}
			void pre_process() {
				dfs(0,-1,g,lvl,pare,dist);
			}
			void dfs(int cur, int pre, ArrayList<edge>[] g, int lvl[], int pare[], int dist[]) {
				for(edge nxt_edge:g[cur]) {
					if(nxt_edge.t!=pre) {
						lvl[nxt_edge.t] = lvl[cur]+1;
						dist[nxt_edge.t] = dist[cur]+nxt_edge.len;
						pare[nxt_edge.t] = cur;
						dfs(nxt_edge.t,cur,g,lvl,pare,dist);
					}
				}				
			}
			public int work(int p, int q) {
				int a = p;
				int b = q;
				while(lvl[p]<lvl[q]) q = pare[q];
				while(lvl[p]>lvl[q]) p = pare[p];			
				while(p!=q){p = pare[p]; q = pare[q];}
				int c = p;
				return dist[a]+dist[b]-dist[c]*2;
			}
		}
		static class lca_binary_lifting{
			int n;
			ArrayList<edge>[] g;
			int lvl[];
			int pare[];
			int dist[];
			int table[][];
			public lca_binary_lifting(int a, ArrayList<edge>[] t) {
				n = a;
				g = t;
				lvl = new int[n];
				pare = new int[n];
				dist = new int[n];				
				table = new int[20][n];				
			}
			void pre_process() {
				dfs(0,-1,g,lvl,pare,dist);
				for(int i=0;i<20;i++) {
					for(int j=0;j<n;j++) {
						if(i==0) table[0][j] = pare[j];
						else table[i][j] = table[i-1][table[i-1][j]];
					}
				}				
			}
			void dfs(int cur, int pre, ArrayList<edge>[] g, int lvl[], int pare[], int dist[]) {
				for(edge nxt_edge:g[cur]) {
					if(nxt_edge.t!=pre) {
						lvl[nxt_edge.t] = lvl[cur]+1;
						dist[nxt_edge.t] = dist[cur]+nxt_edge.len;
						pare[nxt_edge.t] = cur;
						dfs(nxt_edge.t,cur,g,lvl,pare,dist);
					}
				}				
			}			
			public int work(int p, int q) {
				int a = p;
				int b = q;
				if(lvl[p]>lvl[q]) {
					int tmp = p;
					p=q;
					q=tmp;
				}				
				for(int i=19;i>=0;i--) {
					if(lvl[table[i][q]]>=lvl[p]) q=table[i][q];
				}				
				if(p==q) return p;// return dist[a]+dist[b]-dist[p]*2;
				for(int i=19;i>=0;i--) {
					if(table[i][p]!=table[i][q]) {
						p = table[i][p];
						q = table[i][q];
					}
				}
				return table[0][p];
				//return dist[a]+dist[b]-dist[table[0][p]]*2;
			}
		}
		static class lca_sqrt_root{
			int n;
			ArrayList<edge>[] g;
			int lvl[];
			int pare[];
			int dist[];
			int jump[];
			int sz;
			public lca_sqrt_root(int a, ArrayList<edge>[] b) {
				n=a;
				g=b;
				lvl = new int[n];
				pare = new int[n];
				dist = new int[n];
				jump = new int[n];
				sz = (int) Math.sqrt(n);
			}
			void pre_process() {
				dfs(0,-1,g,lvl,pare,dist,jump);
			}
			void dfs(int cur, int pre, ArrayList<edge>[] g, int lvl[], int pare[], int dist[], int[] jump) {
				for(edge nxt_edge:g[cur]) {
					if(nxt_edge.t!=pre) {
						lvl[nxt_edge.t] = lvl[cur]+1;
						dist[nxt_edge.t] = dist[cur]+nxt_edge.len;
						pare[nxt_edge.t] = cur;
						if(lvl[nxt_edge.t]%sz==0) {
							jump[nxt_edge.t] = cur;
						}else {
							jump[nxt_edge.t] = jump[cur];
						}
						dfs(nxt_edge.t,cur,g,lvl,pare,dist,jump);
					}
				}				
			}
			int work(int p, int q) {
				int a = p; int b = q;
				if(lvl[p]>lvl[q]) {
					int tmp = p;
					p=q;
					q=tmp;
				}
				while(jump[p]!=jump[q]) {
					if(lvl[p]>lvl[q]) p = jump[p];
					else q = jump[q];
				}
				while(p!=q) {
					if(lvl[p]>lvl[q]) p = pare[p];
					else q = pare[q];
				}
				return dist[a]+dist[b]-dist[p]*2;
			}
		}
		
//		class edge implements Comparable<edge>{
//			int f,t,len;
//			public edge(int a, int b, int c) {
//				f=a;t=b;len=c;
//			}
//			@Override
//			public int compareTo(edge t) {				
//				return t.len-this.len;
//			}
//		}
	
		class pair implements Comparable<pair>{
			int idx,lvl;
			public pair(int a, int b) {
				idx = a;
				lvl = b;
			}
			@Override
			public int compareTo(pair t) {
				return t.lvl-this.lvl;
			}
		}		
		
		static class lca_RMQ{
			int n;
			ArrayList<edge>[] g;
			int lvl[];
			int dist[];
			int tour[];
			int tour_rank[];
			int first_occ[];
			int c;
			sgt s;
			public lca_RMQ(int a, ArrayList<edge>[] b) {
				n=a;
				g=b;
				c=0;
				lvl = new int[n];
				dist = new int[n];
				tour = new int[2*n];
				tour_rank = new int[2*n];
				first_occ = new int[n];
				Arrays.fill(first_occ, -1);
			}
			void pre_process() {
				tour[c++] = 0;
				dfs(0,-1);
				for(int i=0;i<2*n;i++) {
					tour_rank[i] = lvl[tour[i]];
					if(first_occ[tour[i]]==-1) first_occ[tour[i]] = i;
				}
				s = new sgt(0,2*n,tour_rank);
			}
			void dfs(int cur, int pre) {
				for(edge nxt_edge:g[cur]) {
					if(nxt_edge.t!=pre) {
						lvl[nxt_edge.t] = lvl[cur]+1;
						dist[nxt_edge.t] = dist[cur]+nxt_edge.len;
						tour[c++] = nxt_edge.t;
						dfs(nxt_edge.t,cur);
						tour[c++] = cur;
					}
				}				
			}			
			int work(int p,int q) {			
				int a = Math.max(first_occ[p], first_occ[q]);
				int b = Math.min(first_occ[p], first_occ[q]);
				int idx = s.query_min_idx(b, a+1);
				//Dumper.print(a+" "+b+" "+idx);
				int c = tour[idx];
				return dist[p]+dist[q]-dist[c]*2;
			}
		}		

		  
	    static class sgt{
	    	sgt lt;
	    	sgt rt;
	    	int l,r;
	    	int sum, max, min, lazy;
	    	int min_idx;
	    	public sgt(int L, int R, int arr[]) {
	    		l=L;r=R;
	    		if(l==r-1) {
	    			sum = max = min = arr[l];
	    			lazy = 0;
	    			min_idx = l;
	    			return;
	    		}
	    		lt = new sgt(l, l+r>>1, arr);
	    		rt = new sgt(l+r>>1, r, arr);
	    		pop_up();
	    	}
	    	void pop_up() {
	    		this.sum = lt.sum + rt.sum;
	    		this.max = Math.max(lt.max, rt.max);
	    		this.min = Math.min(lt.min, rt.min);
	    		if(lt.min<rt.min)
	    			this.min_idx = lt.min_idx;
	    		else if(lt.min>rt.min) this.min_idx = rt.min_idx;
	    		else this.min = Math.min(lt.min_idx, rt.min_idx);
	    	}
	    	void push_down() {
	    		if(this.lazy!=0) {
	    			lt.sum+=lazy;
	    			rt.sum+=lazy;
	    			lt.max+=lazy;
	    			lt.min+=lazy;
	    			rt.max+=lazy;
	    			rt.min+=lazy;
	    			lt.lazy+=this.lazy;
	    			rt.lazy+=this.lazy;
	    			this.lazy = 0;
	    		}
	    	}
	    	void change(int L, int R, int v) {
	    		if(R<=l||r<=L) return;
	    		if(L<=l&&r<=R) {
	    			this.max+=v;
	    			this.min+=v;
	    			this.sum+=v*(r-l);
	    			this.lazy+=v;
	    			return;
	    		}
	    		push_down();
	    		lt.change(L, R, v);
	    		rt.change(L, R, v);
	    		pop_up();
	    	}
	    	int query_max(int L, int R) {
	    		if(L<=l&&r<=R) return this.max;
	    		if(r<=L||R<=l) return Integer.MIN_VALUE;
	    		push_down();
	    		return Math.max(lt.query_max(L, R), rt.query_max(L, R));	    		
	    	}
	    	int query_min(int L, int R) {
	    		if(L<=l&&r<=R) return this.min;
	    		if(r<=L||R<=l) return Integer.MAX_VALUE;
	    		push_down();
	    		return Math.min(lt.query_min(L, R), rt.query_min(L, R));	
	    	}
	    	int query_sum(int L, int R) {
	    		if(L<=l&&r<=R) return this.sum;
	    		if(r<=L||R<=l) return 0;
	    		push_down();
	    		return lt.query_sum(L, R) + rt.query_sum(L, R);
	    	}
	    	int query_min_idx(int L, int R) {
	    		if(L<=l&&r<=R) return this.min_idx;
	    		if(r<=L||R<=l) return Integer.MAX_VALUE;
	    		int a = lt.query_min_idx(L, R);
	    		int b = rt.query_min_idx(L, R);
	    		int aa = lt.query_min(L, R);
	    		int bb = rt.query_min(L, R);
	    		if(aa<bb) return a;
	    		else if(aa>bb) return b;	    		
	    		return Math.min(a,b);
	    	}
	    }	   
			    	
		List<List<Integer>> convert(int arr[][]){
			int n = arr.length;
			List<List<Integer>> ret = new ArrayList<>();
			for(int i=0;i<n;i++) {
				ArrayList<Integer> tmp = new ArrayList<Integer>();
				for(int j=0;j<arr[i].length;j++) tmp.add(arr[i][j]);
				ret.add(tmp);
			}
			return ret;
		}
		
		public class TreeNode {
			int val;
			TreeNode left;
			TreeNode right;
			TreeNode(int x) { val = x; }
		}
	
		public int GCD(int a, int b) {
		   if (b==0) return a;
		   return GCD(b,a%b);
		}
		public long GCD(long a, long b) {
			   if (b==0) return a;
			   return GCD(b,a%b);
		}		
	}
	
	
    static class ArrayUtils {
        static final long seed = System.nanoTime();
        static final Random rand = new Random(seed);
 
        public static void sort(int[] a) {
            shuffle(a);
            Arrays.sort(a);
        }
 
        public static void shuffle(int[] a) {
            for (int i = 0; i < a.length; i++) {
                int j = rand.nextInt(i + 1);
                int t = a[i];
                a[i] = a[j];
                a[j] = t;
            }
        }
        
        public static void sort(long[] a) {
            shuffle(a);
            Arrays.sort(a);
        }        
        
        public static void shuffle(long[] a) {
            for (int i = 0; i < a.length; i++) {
                int j = rand.nextInt(i + 1);
                long t = a[i];
                a[i] = a[j];
                a[j] = t;
            }
        }
    }	
	static class BIT{
		int arr[];
		int n;
		public BIT(int a) {
			n=a;
			arr = new int[n];
		}
		int sum(int p) {
			int s=0;
			while(p>0) {
				s+=arr[p];
				p-=p&(-p);
			}
			return s;
		}
		void add(int p, int v) {
			while(p<n) {
				arr[p]+=v;
				p+=p&(-p);
			}
		}
	}
	static class DSU{
		int[] arr;
		int[] sz;
		public DSU(int n) {
			arr = new int[n];
			sz = new int[n];
			for(int i=0;i<n;i++) arr[i] = i;
			Arrays.fill(sz, 1);
		}
		public int find(int a) {
			if(arr[a]!=a) arr[a] = find(arr[a]);
			return arr[a];
		}
		public void union(int a, int b) {
			int x = find(a);
			int y = find(b);
			if(x==y) return;
			arr[y] = x;
			sz[x] += sz[y];
		}
		public int size(int x) {
			return sz[find(x)];
		}
	}	
	static class MinHeap<Key> implements Iterable<Key> {
		private int maxN;
		private int n;
		private int[] pq;
		private int[] qp;
		private Key[] keys;
		private Comparator<Key> comparator;
		
		public MinHeap(int capacity){
			if (capacity < 0) throw new IllegalArgumentException();
			this.maxN = capacity;
			n=0;
			pq = new int[maxN+1];
			qp = new int[maxN+1];
			keys = (Key[]) new Object[capacity+1];
			Arrays.fill(qp, -1);
		}
		
		public MinHeap(int capacity, Comparator<Key> c){
			if (capacity < 0) throw new IllegalArgumentException();
			this.maxN = capacity;
			n=0;
			pq = new int[maxN+1];
			qp = new int[maxN+1];
			keys = (Key[]) new Object[capacity+1];
			Arrays.fill(qp, -1);
			comparator = c;
		}			
		public boolean isEmpty()	{ return n==0; 	}
		public int size()			{ return n;		}
		public boolean contains(int i) {
	        if (i < 0 || i >= maxN) throw new IllegalArgumentException();
	        return qp[i] != -1;
		}	
		public int peekIdx() {
	        if (n == 0) throw new NoSuchElementException("Priority queue underflow");
	        return pq[1];		
		}	
		public Key peek(){
			if(isEmpty()) throw new NoSuchElementException("Priority queue underflow");
			return keys[pq[1]];
		}
		public int poll(){
			if(isEmpty()) throw new NoSuchElementException("Priority queue underflow");
			int min = pq[1];
			exch(1,n--);
			down(1);
			assert min==pq[n+1];
			qp[min] = -1;
			keys[min] = null;		
			pq[n+1] = -1;
			return min;
		}
	    public void update(int i, Key key) {
	        if (i < 0 || i >= maxN) throw new IllegalArgumentException();
	        if (!contains(i)) {
	        	this.add(i, key);
	        }else {
	        	keys[i] = key;
	        	up(qp[i]);
	        	down(qp[i]);
	        }
	    }	
		private void add(int i, Key x){
	        if (i < 0 || i >= maxN) throw new IllegalArgumentException();
	        if (contains(i)) throw new IllegalArgumentException("index is already in the priority queue");
	        n++;
	        qp[i] = n;
	        pq[n] = i;
	        keys[i] = x;
	        up(n);
		}
		private void up(int k){
			while(k>1&&less(k,k/2)){
				exch(k,k/2);
				k/=2;
			}
		}
		private void down(int k){
			while(2*k<=n){
				int j=2*k;
				if(j<n&&less(j+1,j)) j++;
				if(less(k,j)) break;
				exch(k,j);
				k=j;
			}
		}
		
		public boolean less(int i, int j){
	        if (comparator == null) {
	            return ((Comparable<Key>) keys[pq[i]]).compareTo(keys[pq[j]]) < 0;
	        }
	        else {
	            return comparator.compare(keys[pq[i]], keys[pq[j]]) < 0;
	        }
		}
 
		public void exch(int i, int j){
	        int swap = pq[i];
	        pq[i] = pq[j];
	        pq[j] = swap;
	        qp[pq[i]] = i;
	        qp[pq[j]] = j;
		}
 
		@Override
		public Iterator<Key> iterator() {
			// TODO Auto-generated method stub
			return null;
		}
	}  	
 
    private static class InputReader
    {
        private InputStream stream;
        private byte[] buf = new byte[1024];
        private int zcurChar;
        private int znumChars;
        private SpaceCharFilter filter;
 
        public InputReader(InputStream stream)
        {
            this.stream = stream;
        }
 
        public int read()
        {
            if (znumChars == -1)
                throw new InputMismatchException();
            if (zcurChar >= znumChars)
            {
            	zcurChar = 0;
                try
                {
                    znumChars = stream.read(buf);
                } catch (IOException e)
                {
                    throw new InputMismatchException();
                }
                if (znumChars <= 0)
                    return -1;
            }
            return buf[zcurChar++];
        }
 
        public int nextInt()
        {
            int c = read();
            while (isSpaceChar(c))
                c = read();
            int sgn = 1;
            if (c == '-')
            {
                sgn = -1;
                c = read();
            }
            int res = 0;
            do
            {
                if (c < '0' || c > '9')
                    throw new InputMismatchException();
                res *= 10;
                res += c - '0';
                c = read();
            } while (!isSpaceChar(c));
            return res * sgn;
        }
 
        public String nextString()
        {
            int c = read();
            while (isSpaceChar(c))
                c = read();
            StringBuilder res = new StringBuilder();
            do
            {
                res.appendCodePoint(c);
                c = read();
            } while (!isSpaceChar(c));
            return res.toString();
        }
 
        public double nextDouble()
        {
            int c = read();
            while (isSpaceChar(c))
                c = read();
            int sgn = 1;
            if (c == '-')
            {
                sgn = -1;
                c = read();
            }
            double res = 0;
            while (!isSpaceChar(c) && c != '.')
            {
                if (c == 'e' || c == 'E')
                    return res * Math.pow(10, nextInt());
                if (c < '0' || c > '9')
                    throw new InputMismatchException();
                res *= 10;
                res += c - '0';
                c = read();
            }
            if (c == '.')
            {
                c = read();
                double m = 1;
                while (!isSpaceChar(c))
                {
                    if (c == 'e' || c == 'E')
                        return res * Math.pow(10, nextInt());
                    if (c < '0' || c > '9')
                        throw new InputMismatchException();
                    m /= 10;
                    res += (c - '0') * m;
                    c = read();
                }
            }
            return res * sgn;
        }
 
        public long nextLong()
        {
            int c = read();
            while (isSpaceChar(c))
                c = read();
            int sgn = 1;
            if (c == '-')
            {
                sgn = -1;
                c = read();
            }
            long res = 0;
            do
            {
                if (c < '0' || c > '9')
                    throw new InputMismatchException();
                res *= 10;
                res += c - '0';
                c = read();
            } while (!isSpaceChar(c));
            return res * sgn;
        }
 
        public boolean isSpaceChar(int c)
        {
            if (filter != null)
                return filter.isSpaceChar(c);
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }
 
        public String next()
        {
            return nextString();
        }
 
        public interface SpaceCharFilter
        {
            public boolean isSpaceChar(int ch);
        }
        public int[] readIntArray(int n) {
            int[] ret = new int[n];
            for (int i = 0; i < n; i++) {
                ret[i] = nextInt();
            }
            return ret;
        }        
    }    
 
	static class Dumper {
		static void print_int_arr(int[] arr) {
			for (int i = 0; i < arr.length; i++) {
				System.out.print(arr[i] + " ");
			}
			System.out.println();
			System.out.println("---------------------");
		}
 
		static void print_char_arr(char[] arr) {
			for (int i = 0; i < arr.length; i++) {
				System.out.print(arr[i] + " ");
			}
			System.out.println();
			System.out.println("---------------------");
		}
 
		static void print_double_arr(double[] arr) {
			for (int i = 0; i < arr.length; i++) {
				System.out.print(arr[i] + " ");
			}
			System.out.println();
			System.out.println("---------------------");
		}
 
		static void print_2d_arr(int[][] arr, int x, int y) {
			for (int i = 0; i < x; i++) {
				for (int j = 0; j < y; j++) {
					System.out.print(arr[i][j] + " ");
				}
				System.out.println();
			}
			System.out.println();
			System.out.println("---------------------");
		}
 
		static void print_2d_arr(boolean[][] arr, int x, int y) {
			for (int i = 0; i < x; i++) {
				for (int j = 0; j < y; j++) {
					System.out.print(arr[i][j] + " ");
				}
				System.out.println();
			}
			System.out.println();
			System.out.println("---------------------");
		}
 
		static void print(Object o) {
			System.out.println(o.toString());
		}
 
		static void getc() {
			System.out.println("here");
		}
	}
}