Problem: Ujan has a lot of numbers in his boxes. He likes order and balance, so he decided to reorder the numbers.

There are k boxes numbered from 1 to k. The i-th box contains n_i integer numbers. The integers can be negative. All of the integers are distinct.

Ujan is lazy, so he will do the following reordering of the numbers exactly once. He will pick a single integer from each of the boxes, k integers in total. Then he will insert the chosen numbers — one integer in each of the boxes, so that the number of integers in each box is the same as in the beginning. Note that he may also insert an integer he picked from a box back into the same box.

Ujan will be happy if the sum of the integers in each box is the same. Can he achieve this and make the boxes perfectly balanced, like all things should be?

Input

The first line contains a single integer k (1 ≤ k ≤ 15), the number of boxes. 

The i-th of the next k lines first contains a single integer n_i (1 ≤ n_i ≤ 5 000), the number of integers in box i. Then the same line contains n_i integers a_{i,1}, …, a_{i,n_i} (|a_{i,j}| ≤ 10^9), the integers in the i-th box. 

It is guaranteed that all a_{i,j} are distinct.

Output

If Ujan cannot achieve his goal, output "No" in a single line. Otherwise in the first line output "Yes", and then output k lines. The i-th of these lines should contain two integers c_i and p_i. This means that Ujan should pick the integer c_i from the i-th box and place it in the p_i-th box afterwards.

If there are multiple solutions, output any of those.

You can print each letter in any case (upper or lower).

Examples

Input


4
3 1 7 4
2 3 2
2 8 5
1 10


Output


Yes
7 2
2 3
5 1
10 4


Input


2
2 3 -2
2 -1 5


Output


No


Input


2
2 -10 10
2 0 -20


Output


Yes
-10 2
-20 1

Note

In the first sample, Ujan can put the number 7 in the 2nd box, the number 2 in the 3rd box, the number 5 in the 1st box and keep the number 10 in the same 4th box. Then the boxes will contain numbers \{1,5,4\}, \{3, 7\}, \{8,2\} and \{10\}. The sum in each box then is equal to 10.

In the second sample, it is not possible to pick and redistribute the numbers in the required way.

In the third sample, one can swap the numbers -20 and -10, making the sum in each box equal to -10.

Solution (in JAVA):

Solution:
import java.io.IOException;
import java.io.PrintWriter;
import java.util.ArrayDeque;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.NoSuchElementException;
import java.util.Objects;
import java.util.PriorityQueue;
import java.util.Queue;
import java.util.Random;
import java.util.Scanner;
import java.util.TreeMap;
import java.util.TreeSet;
import java.util.function.BiFunction;

public class Main{
	static Scanner scn = new Scanner(System.in);
	static FastScanner sc = new FastScanner();
	static Mathplus mp = new Mathplus();
	static PrintWriter ot = new PrintWriter(System.out);
	static Random rand = new Random();
	static int mod = 1000000007;
	static long inf = (long)1e17;
	static int[] dx = {0,1,0,-1};
	static int[] dy = {1,0,-1,0};
	static int max;
	public static void main(String[] args) {	
		
		int N = sc.nextInt();
		ArrayList<Integer>[] l = new ArrayList[N];
		HashMap<Long,Integer> map = new HashMap<Long,Integer>();
		long sum = 0;
		long[] s = new long[N];
		for(int i=0;i<N;i++) {
			l[i] = new ArrayList<Integer>();
			int n = sc.nextInt();
			for(int j=0;j<n;j++) {
				int a = sc.nextInt();
				map.put((long)a, i);
				s[i] += a;
				sum += a;
				l[i].add(a);
			}
		}
		if(Math.abs(sum)%N!=0) {
			System.out.println("NO");
			return;
		}
		long make = sum/N;
		boolean[] dp = new boolean[(1<<N)];
		int[] first = new int[(1<<N)];
		int[] bef = new int[(1<<N)];

		Arrays.fill(first,mod);
		for(int i=0;i<N;i++) {
			for(int a:l[i]) {
				int used = 0;
				boolean f = true;
				long now = a;
				int see = i;
				while(true) {
					long next = make-(s[see]-now);
					
					if(next==a) {
						break;
					}
					if(!map.containsKey(next)) {
						f = false;
						break;
					}else {
						int k = map.get(next);
						if(mp.contains(used,k)&&k!=i) {
							f = false;
							break;
						}else {
							used = mp.bitadd(used,k);
							now = next;
							see = k;
						}
					}
				}
				if(f) {
					dp[mp.bitadd(used,i)] = true;
					first[mp.bitadd(used,i)] = a;
				}
			}
		}
		
		dp[0] = true;
		for(int i=1;i<(1<<N);i++) {
			for(int j=i;j>0;j=(j-1)&i) {
				if(dp[i^j]&&dp[j]) {
					dp[i]=true;
					bef[i] = j;
				}
			}
		}
		
		if(!dp[(1<<N)-1]) {
			System.out.println("NO");
		}else {
			System.out.println("YES");
			ArrayDeque<Integer> q = new ArrayDeque<Integer>();
			int[] ans1 = new int[N];
			int[] ans2 = new int[N];
			q.add((1<<N)-1);
			while(!q.isEmpty()) {
				int Q = q.poll();
				if(first[Q]==mod) {
					q.add(bef[Q]);
					q.add(Q^bef[Q]);
				}else {
					
					int a = first[Q];
					long now = a;
					int befo = map.get((long)a);
					while(true) {
						long next = make-(s[befo]-now);
						if(next==a) {
							int k = map.get(next);
							ans1[k] = (int)next;
							ans2[k] = befo;
							break;
						}
						int k = map.get(next);
						ans1[k] = (int)next;
						ans2[k] = befo;
						now = next;
						befo = k;
					}
				}
			}
			for(int i=0;i<N;i++) {
				System.out.println(ans1[i]+" "+(ans2[i]+1));
			}
			
		}
		
		
		
		
	
	
	}
	
}
class GridGraph extends Graph{
	
	int N;
	int M;
	String[] S;
	HashMap<Character,Integer> map;
	GridGraph(int n,int m,String[] s,char[] c){
		super(n*m);
		N = n;
		M = m;
		S = s;
		map = new HashMap<Character,Integer>();
		for(int i=0;i<n-1;i++) {
			for(int j=0;j<m;j++) {
				if(S[i].charAt(j)!='#'&&S[i+1].charAt(j)!='#') {
					addEdge(toint(i,j),toint(i+1,j));
					addEdge(toint(i+1,j),toint(i,j));
				}
			}
		}
		for(int i=0;i<n;i++) {
			for(int j=0;j<m-1;j++) {
				if(S[i].charAt(j)!='#'&&S[i].charAt(j+1)!='#') {
					addEdge(toint(i,j),toint(i,j+1));
					addEdge(toint(i,j+1),toint(i,j));
					
				}
			}
		}
		for(int i=0;i<n;i++) {
			for(int j=0;j<m;j++) {
				for(int k=0;k<c.length;k++) {
					if(S[i].charAt(j)==c[k])map.put(c[k],toint(i,j));
				}
			}
		}
	}
	
	int toint(int i,int j) {
		return i*M+j;
	}
	
}

class BetterGridGraph{
	int N;
	int M;
	char[][] S;
	HashMap<Character,Integer> map;
	int[] dx = {0,1,0,-1};
	int[] dy = {1,0,-1,0};
	char w;
	char b = '#';
	BetterGridGraph(int n,int m,String[] s,char[] c){
		
		N = n;
		M = m;
		for(int i=0;i<s.length;i++) {
			S[i] = s[i].toCharArray();
		}
		map = new HashMap<Character,Integer>();
		for(int i=0;i<n;i++) {
			for(int j=0;j<m;j++) {
				for(int k=0;k<c.length;k++) {
					if(S[i][j]==c[k])map.put(c[k],toint(i,j));
				}
			}
		}
	}
	BetterGridGraph(int n,int m,char[][] s,char[] c){
		
		N = n;
		M = m;
		S = s;
		map = new HashMap<Character,Integer>();
		for(int i=0;i<n;i++) {
			for(int j=0;j<m;j++) {
				for(int k=0;k<c.length;k++) {
					if(S[i][j]==c[k])map.put(c[k],toint(i,j));
				}
			}
		}
	}
	
	BetterGridGraph(int n,int m,String[] s,char[] c,char W,char B){
		
		N = n;
		M = m;
		for(int i=0;i<s.length;i++) {
			S[i] = s[i].toCharArray();
		}
		w = W;
		b = B;
		map = new HashMap<Character,Integer>();
		for(int i=0;i<n;i++) {
			for(int j=0;j<m;j++) {
				for(int k=0;k<c.length;k++) {
					if(S[i][j]==c[k])map.put(c[k],toint(i,j));
				}
			}
		}
	}
	BetterGridGraph(int n,int m,char[][] s,char[] c,char W,char B){
		
		N = n;
		M = m;
		S = s;
		w = W;
		b = B;
		map = new HashMap<Character,Integer>();
		for(int i=0;i<n;i++) {
			for(int j=0;j<m;j++) {
				for(int k=0;k<c.length;k++) {
					if(S[i][j]==c[k])map.put(c[k],toint(i,j));
				}
			}
		}
	}
	
	int toint(int i,int j) {
		return i*M+j;
	}
	
	int[] bfs(char C) {
		int s = map.get(C);
		int[] L = new int[N*M];
		for(int i=0;i<N*M;i++){
			L[i] = -1;
		}
		L[s] = 0;
		ArrayDeque<Integer> Q = new ArrayDeque<Integer>();
		Q.add(s);
		Range X = new Range(0,N-1);
		Range Y = new Range(0,M-1);
		while(!Q.isEmpty()){
			int v = Q.poll();
			for(int i=0;i<4;i++){
				int x = v/M;
				int y = v%M;
				int nx = x+dx[i];
				int ny = y+dy[i];
				if(X.isIn(nx)&&Y.isIn(ny)&&S[nx][ny]!=b) {
					int w = toint(nx,ny);
					if(L[w]==-1){
						L[w] = L[v] + 1;
						Q.add(w);
					}
				}
			}
		}
		return L;
	}
	
	int[][] bfs2(char C,int K){
		int s = map.get(C);
		int[][] L = new int[N*M][K+1];
		for(int i=0;i<N*M;i++){
			for(int j=0;j<=K;j++)
			L[i][j] = 1000000007;
		}
		L[s][0] = 0;
		ArrayDeque<IntIntPair> Q = new ArrayDeque<IntIntPair>();
		Q.add(new IntIntPair(s,0));
		Range X = new Range(0,N-1);
		Range Y = new Range(0,M-1);
		while(!Q.isEmpty()){
			IntIntPair v = Q.poll();
			for(int i=0;i<4;i++){
				int x = v.a/M;
				int y = v.a%M;
				int h = v.b;
				int nx = x+dx[i];
				int ny = y+dy[i];
				if(X.isIn(nx)&&Y.isIn(ny)&&S[nx][ny]!=b) {
					int ni = toint(nx,ny);
					int nh = S[nx][ny]==w?h+1:h;
					if(nh>K) continue;
					if(L[ni][nh]==1000000007){
						L[ni][nh] = L[v.a][h] + 1;
						Q.add(new IntIntPair(ni,nh));
					}
				}
			}
		}
		for(int i=0;i<N*M;i++) {
			for(int j=1;j<=K;j++) {
				L[i][j] = Math.min(L[i][j],L[i][j-1]);
			}
		}
		return L;
	}
}

class StringManager{
	ArrayList<Character> S;
	static Mathplus mp;
	static boolean calced;
	static int base;
	static long baserev;
	ArrayList<Long> l;
	StringManager(String s){
		S = new ArrayList<Character>();
		for(int i=0;i<s.length();i++) {
			S.add(s.charAt(i));
		}
		if(!calced) {
			calced = true;
			mp = new Mathplus();
			base = 1000003;
			baserev = mp.rev(base);
			mp.buildpow(base,1000050);
			mp.buildrevpow((int) baserev,1000050);

		}
		l = new ArrayList<Long>();

		l.add((long)S.get(0));

		for(int i=1;i<S.size();i++) {
			char c = S.get(i);
			l.add((l.get(i-1) + mp.pow[i] * c)%mp.mod);

		}
	}
	void add(char C){
		int i = S.size();
		S.add(C);
		l.add((l.get(i-1) + mp.pow[i] * C)%mp.mod);
	}
	long gethash(int le,int ri) {
		long res = l.get(ri);
		if(le!=0) {
			res -= l.get(le-1);
			res += mp.mod;
			res %= mp.mod;
			res *= mp.revpow[le];
			res %= mp.mod;
		}
		return res;
	}


}


class Trie{
	int nodenumber = 1;
	ArrayList<TrieNode> l;
	Trie(){
		l = new ArrayList<TrieNode>();
		l.add(new TrieNode());
	}

	void add(String S,int W){
		int now = 0;
		for(int i=0;i<S.length();i++) {
			TrieNode n = l.get(now);
			char c = S.charAt(i);
			if(n.Exist[c-'a']!=-1) {
				now = n.Exist[c-'a'];
			}else {
				l.add(new TrieNode());
				n.Exist[c-'a'] = nodenumber;
				now = nodenumber;
				nodenumber++;
			}
		}
		l.get(now).weight = W;
	}

	void find(String S,int i,int[] dp) {
		int now = 0;
		dp[i+1] = Math.max(dp[i],dp[i+1]);
		for(int j=0;;j++) {
			TrieNode n = l.get(now);
			dp[i+j] = Math.max(dp[i+j],dp[i]+n.weight);
			int slook = i+j;
			if(slook>=S.length())return;
			char c = S.charAt(slook);
			if(n.Exist[c-'a']==-1)return;
			now = n.Exist[c-'a'];
		}
	}
}

class TrieNode{

	int[] Exist = new int[26];
	int weight = 0;
	TrieNode(){
		for(int i=0;i<26;i++) {
			Exist[i] = -1;
		}
	}
}

class SizeComparator implements Comparator<Edge>{
	int[] size;
	SizeComparator(int[] s) {
		size = s;
	}

	public int compare(Edge o1, Edge o2) {
		return size[o1.to]-size[o2.to];

	}

}

class ConvexHullTrick {
	long[] A, B;
	int len;

	public ConvexHullTrick(int n) {
		A = new long[n];
		B = new long[n];
	}

	private boolean check(long a, long b) {
		return (B[len - 2] - B[len - 1]) * (a - A[len - 1]) >= (B[len - 1] - b) * (A[len - 1] - A[len - 2]);
	}

	public void add(long a, long b) {
		while (len >= 2 && check(a, b)) {
			len--;
		}
		A[len] = a;
		B[len] = b;
		len++;
	}

	public long query(long x) {
		int l = -1, r = len - 1;
		while (r - l > 1) {
			int mid = (r + l) / 2;
			if (get(mid,x)>=get(mid+1,x)) {
				l = mid;
			} else {
				r = mid;
			}
		}
		return get(r,x);
	}

	private long get(int k, long x) {
		return A[k] * x + B[k];
	}
}

class Range{
	int l;
	int r;
	int length;
	Range(int L,int R){
		l = L;
		r = R;
		length = R-L+1;
	}

	boolean isIn(int x) {
		return (l<=x&&x<=r);

	}
}
class LeftComparator implements Comparator<Range>{
	public int compare(Range P, Range Q) {
		return P.l-Q.l;
	}
}
class RightComparator implements Comparator<Range>{
	public int compare(Range P, Range Q) {
		return P.r-Q.r;
	}
}
class LengthComparator implements Comparator<Range>{
	public int compare(Range P, Range Q) {
		return P.length-Q.length;
	}
}
class SegmentTree<T,E>{
	int N;
	BiFunction<T,T,T> f;
	BiFunction<T,E,T> g;
	T d1;
	ArrayList<T> dat;
	SegmentTree(BiFunction<T,T,T> F,BiFunction<T,E,T> G,T D1,T[] v){
		int n = v.length;
		f = F;
		g = G;
		d1 = D1;
		init(n);
		build(v);
	}


	void init(int n) {
		N = 1;
		while(N<n)N*=2;
		dat = new ArrayList<T>();
	}

	void build(T[] v) {
		for(int i=0;i<2*N;i++) {
			dat.add(d1);
		}
		for(int i=0;i<v.length;i++) {
			dat.set(N+i-1,v[i]);
		}
		for(int i=N-2;i>=0;i--) {
			dat.set(i,f.apply(dat.get(i*2+1),dat.get(i*2+2)));
		}
	}

	void update(int k,E a) {
		k += N-1;
		dat.set(k,g.apply(dat.get(k),a));
		while(k>0){
			k = (k-1)/2;
			dat.set(k,f.apply(dat.get(k*2+1),dat.get(k*2+2)));
		}
	}

	T query(int a,int b, int k, int l ,int r) {
		if(r<=a||b<=l) return d1;
		if(a<=l&&r<=b) return dat.get(k);
		T vl = query(a,b,k*2+1,l,(l+r)/2);
		T vr = query(a,b,k*2+2,(l+r)/2,r);
		return f.apply(vl, vr);
	}
	T query(int a,int b){
		return query(a,b,0,0,N);
	}

}

class LazySegmentTree<T,E> extends SegmentTree<T,E>{
	BiFunction<E,E,E> h;
	BiFunction<E,Integer,E> p = (E a,Integer b) ->{return a;};
	E d0;
	ArrayList<E> laz;
	LazySegmentTree(BiFunction<T,T,T> F,BiFunction<T,E,T> G,BiFunction<E,E,E> H,T D1,E D0,T[] v){
		super(F,G,D1,v);
		int n = v.length;
		h = H;
		d0 = D0;
		Init(n);
	}
	void build() {

	}
	void Init(int n){
		laz = new ArrayList<E>();
		for(int i=0;i<2*N;i++) {
			laz.add(d0);
		}
	}

	void eval(int len,int k) {
		if(laz.get(k).equals(d0)) return;
		if(k*2+1<N*2-1) {
			laz.set(k*2+1,h.apply(laz.get(k*2+1),laz.get(k)));
			laz.set(k*2+2,h.apply(laz.get(k*2+2),laz.get(k)));
		}
		dat.set(k,g.apply(dat.get(k), p.apply(laz.get(k), len)));
		laz.set(k,d0);
	}

	T update(int a,int b,E x,int k,int l,int r) {
		eval(r-l,k);
		if(r<=a||b<=l) {
			return dat.get(k);
		}
		if(a<=l&&r<=b) {
			laz.set(k,h.apply(laz.get(k),x));
			return g.apply(dat.get(k),p.apply(laz.get(k),r-l));
		}
		T vl = update(a,b,x,k*2+1,l,(l+r)/2);
		T vr = update(a,b,x,k*2+2,(l+r)/2,r);
		dat.set(k,f.apply(vl,vr));
		return dat.get(k);

	}

	T update(int a,int b,E x) {
		return update(a,b,x,0,0,N);
	}

	T query(int a,int b,int k,int l,int r) {
		eval(r-l,k);
		if(r<=a||b<=l) return d1;
		if(a<=l&&r<=b) return dat.get(k);

		T vl = query(a,b,k*2+1,l,(l+r)/2);
		T vr = query(a,b,k*2+2,(l+r)/2,r);
		return f.apply(vl, vr);
	}

	T query(int a,int b){
		return query(a,b,0,0,N);
	}

}

class AddSumSegmentTree{
	int N;
	int d1;
	ArrayList<Integer> dat;
	AddSumSegmentTree(int[] v){
		int n = v.length;
		init(n);
		build(v);
	}

	void init(int n) {
		N = 1;
		while(N<n)N*=2;
		dat = new ArrayList<Integer>();
	}

	void build(int[] v) {
		for(int i=0;i<2*N;i++) {
			dat.add(d1);
		}
		for(int i=0;i<v.length;i++) {
			dat.set(N+i-1,v[i]);
		}
		for(int i=N-2;i>=0;i--) {
			dat.set(i,dat.get(i*2+1)+dat.get(i*2+2));
		}
	}

	void update(int k,int a) {
		k += N-1;
		dat.set(k,dat.get(k)+a);
		while(k>0){
			k = (k-1)/2;
			dat.set(k,dat.get(k*2+1)+dat.get(k*2+2));
		}
	}

	int query(int a,int b, int k, int l ,int r) {
		if(r<=a||b<=l) return d1;
		if(a<=l&&r<=b) return dat.get(k);
		int vl = query(a,b,k*2+1,l,(l+r)/2);
		int vr = query(a,b,k*2+2,(l+r)/2,r);
		return vl+vr;
	}
	int query(int a,int b){
		return query(a,b,0,0,N);
	}
}
class AddSumLazySegmentTree {
	int N;
	long[] dat;
	long[] laz;
	AddSumLazySegmentTree(long[] v){
		init(v.length);

		for(int i=0;i<v.length;i++) {
			dat[N+i-1]=v[i];
		}
		for(int i=N-2;i>=0;i--) {
			dat[i]=dat[i*2+1]+dat[i*2+2];
		}
	}

	void init(int n) {
		N = 1;
		while(N<n)N*=2;
		dat = new long[2*N];
		laz = new long[2*N];
	}


	void eval(int len,int k) {
		if(laz[k]==0) return;
		if(k*2+1<N*2-1) {
			laz[k*2+1] += laz[k];
			laz[k*2+2] += laz[k];
		}
		dat[k] += laz[k] * len;
		laz[k] = 0;
	}

	long update(int a,int b,long x,int k,int l,int r) {
		eval(r-l,k);
		if(r<=a||b<=l) {
			return dat[k];
		}
		if(a<=l&&r<=b) {
			laz[k] += x;
			return dat[k]+laz[k]*(r-l);
		}
		long vl = update(a,b,x,k*2+1,l,(l+r)/2);
		long vr = update(a,b,x,k*2+2,(l+r)/2,r);
		return dat[k] = vl+vr;


	}

	long update(int a,int b,long x) {
		return update(a,b,x,0,0,N);
	}

	long query(int a,int b,int k,int l,int r) {
		eval(r-l,k);
		if(r<=a||b<=l) return 0;
		if(a<=l&&r<=b) return dat[k];

		long vl = query(a,b,k*2+1,l,(l+r)/2);
		long vr = query(a,b,k*2+2,(l+r)/2,r);
		return vl+vr;
	}

	long query(int a,int b){
		return query(a,b,0,0,N);
	}

}

class BinaryIndexedTree{
	int[] val;
	BinaryIndexedTree(int N){
		val = new int[N+1];
	}
	long sum(int i) {
		if(i==0)return 0;
		long s = 0;
		while(i>0) {
			s += val[i];
			i -= i & (-i);
		}
		return s;
	}
	void add(int x,int i) {
		if(i==0)return;
		while(i<val.length){
			val[i] += x;
			i += i & (-i);
		}
	}
}

class UnionFindTree {
	int[] root;
	int[] rank;
	int[] size;
	int[] edge;
	int num;
	UnionFindTree(int N){
		root = new int[N];
		rank = new int[N];
		size = new int[N];
		edge = new int[N];
		num = N;
		for(int i=0;i<N;i++){
			root[i] = i;
			size[i] = 1;
		}
	}
	public boolean isRoot(int x) {
		return x==find(x);
	}
	public int extraEdge(int x) {
		int r = find(x);
		return edge[r] - size[r] + 1;
	}
	public int find(int x){
		if(root[x]==x){
			return x;
		}else{
			return find(root[x]);
		}
	}

	public void unite(int x,int y){
		x = find(x);
		y = find(y);
		if(x==y){
			edge[x]++;
			return;
		}else{
			num--;
			if(rank[x]<rank[y]){
				root[x] = y;
				size[y] += size[x];
				edge[y] += edge[x]+1;
			}else{
				root[y] = x;
				size[x] += size[y];
				edge[x] += edge[y]+1;
				if(rank[x]==rank[y]){
					rank[x]++;
				}
			}
		}
	}

	public boolean same(int x,int y){
		return find(x)==find(y);
	}

}

class ParticalEternalLastingUnionFindTree extends UnionFindTree{
	int[] time;
	int now;
	ParticalEternalLastingUnionFindTree(int N){
		super(N);
		time = new int[N];
		for(int i=0;i<N;i++) {
			time[i] = 1000000007;
		}
	}

	public int find(int t,int i) {
		if(time[i]>t) {
			return i;
		}else {
			return find(t,root[i]);
		}
	}

	public void unite(int x,int y,int t) {
		now = t;
		x = find(t,x);
		y = find(t,y);
		if(x==y)return;
		if(rank[x]<rank[y]){
			root[x] = y;
			size[y] += size[x];
			time[x] = t;
		}else{
			root[y] = x;
			size[x] += size[y];
			if(rank[x]==rank[y]){
				rank[x]++;
			}
			time[y] = t;
		}
	}

	public int sametime(int x,int y) {
		if(find(now,x)!=find(now,y)) return -1;
		int ok = now;
		int ng = 0;
		while(ok-ng>1) {
			int mid = (ok+ng)/2;
			if(find(mid,x)==find(mid,y)) {
				ok = mid;
			}else {
				ng = mid;
			}
		}
		return ok;
	}


}
class Graph {
	ArrayList<Edge>[] list;
	int size;
	TreeSet<LinkEdge> Edges = new TreeSet<LinkEdge>(new LinkEdgeComparator());
	
	@SuppressWarnings("unchecked")
	Graph(int N){
		size = N;
		list = new ArrayList[N];
		for(int i=0;i<N;i++){
			list[i] = new ArrayList<Edge>();
		}
	}

	void addEdge(int a,int b){
		list[a].add(new Edge(b,1));
	}

	void addWeightedEdge(int a,int b,long c){
		list[a].add(new Edge(b,c));
	}

	void addEgdes(int[] a,int[] b){

		for(int i=0;i<a.length;i++){
			list[a[i]].add(new Edge(b[i],1));
		}
	}

	void addWeighterEdges(int[] a ,int[] b ,int[] c){
		for(int i=0;i<a.length;i++){
			list[a[i]].add(new Edge(b[i],c[i]));
		}
	}

	long[][] bfs(int s){
		long[][] L = new long[2][size];
		for(int i=0;i<size;i++){
			L[0][i] = -1;
			L[1][i] = -1;
		}
		L[0][s] = 0;
		ArrayDeque<Integer> Q = new ArrayDeque<Integer>();
		Q.add(s);

		while(!Q.isEmpty()){
			int v = Q.poll();
			for(Edge e:list[v]){
				int w = e.to;
				long c = e.cost;
				if(L[0][w]==-1){
					L[0][w] = L[0][v] + c;
					L[1][w] = v; 
					Q.add(w);
				}
			}
		}
		return L;
	}
	long[] bfs2(int[] d,int s){
		long[] L = new long[size];
		for(int i=0;i<size;i++){
			L[i] = -1;
		}
		int p = 0;
		L[s] = 0;
		d[s] = p;
		p++;
		ArrayDeque<Integer> Q = new ArrayDeque<Integer>();
		Q.add(s);

		while(!Q.isEmpty()){
			int v = Q.poll();
			for(Edge e:list[v]){
				int w = e.to;
				long c = e.cost;
				if(L[w]==-1){
					d[w] = p;
					p++;
					L[w] = L[v] + c;
					Q.add(w);
				}
			}
		}
		return L;
	}

	int[] isTwoColor(){
		int[] L = new int[size];
		for(int i=0;i<size;i++){
			L[i] = -1;
		}
		L[0] = 0;
		ArrayDeque<Integer> Q = new ArrayDeque<Integer>();
		Q.add(0);

		while(!Q.isEmpty()){
			int v = Q.poll();
			for(Edge e:list[v]){
				int w = e.to;
				if(L[w]==-1){
					L[w] = 1-L[v];
					Q.add(w);
				}else{
					if(L[v]+L[w]!=1){
						L[0] = -2;
					}
				}
			}
		}
		return L;
	}

	long[] dijkstra(int s){
		long[] L = new long[size];
		for(int i=0;i<size;i++){
			L[i] = -1;
		}
		int[] v = new int[size];
		L[s] = 0;
		PriorityQueue<LongIntPair> Q = new PriorityQueue<LongIntPair>(new LongIntSampleComparator());
		Q.add(new LongIntPair(0,s));

		while(!Q.isEmpty()){

			LongIntPair C = Q.poll();
			
			if(v[C.b]==0){
				L[C.b] = C.a;
				v[C.b] = 1;
				for(Edge D:list[C.b]) {
					if(L[D.to]==-1||L[D.to]>L[C.b]+D.cost) {
						L[D.to]=L[C.b]+D.cost;
						Q.add(new LongIntPair(L[C.b]+D.cost,D.to));
					}
				
				}
			}
		}
		return L;
	}

	ArrayList<Graph> makeapart(){
		ArrayList<Graph> ans = new ArrayList<Graph>();
		boolean[] b = new boolean[size];
		int[] num = new int[size];
		for(int i=0;i<size;i++){
			if(b[i])continue;
			int sz = 0;
			ArrayList<Integer> l = new ArrayList<Integer>();
			ArrayDeque<Integer> Q = new ArrayDeque<Integer>();
			Q.add(i);
			b[i] = true;
			while(!Q.isEmpty()){
				int v = Q.poll();
				num[v] = sz;
				sz++;
				l.add(v);
				for(Edge e:list[v]){
					if(!b[e.to]){
						Q.add(e.to);
						b[e.to] = true;
					}
				}
			}
			Graph H = new Graph(sz);
			for(int e:l){
				for(Edge E:list[e]){
					H.addWeightedEdge(num[e],num[E.to],E.cost);
				}
			}
			ans.add(H);

		}
		return ans;
	}

	long[] bellmanFord(int s) {
		long inf = 1000000000;
		inf *= inf;
		long[] d = new long[size];
		boolean[] n = new boolean[size];
		d[s] = 0;
		for(int i=1;i<size;i++){
			d[i] = inf;
			d[i] *= d[i];
		}
		for(int i=0;i<size-1;i++){
			for(int j=0;j<size;j++){
				for(Edge E:list[j]){
					if(d[j]!=inf&&d[E.to]>d[j]+E.cost){
						d[E.to]=d[j]+E.cost;
					}
				}
			}
		}
		for(int i=0;i<size;i++){
			for(int j=0;j<size;j++){
				for(Edge e:list[j]){
					if(d[j]==inf) continue;
					if(d[e.to]>d[j]+e.cost) {
						d[e.to]=d[j]+e.cost;
						n[e.to] = true;
					}
					if(n[j])n[e.to] = true;
				}
			}
		}
		for(int i=0;i<size;i++) {
			if(n[i])d[i] = inf;
		}
		return d;
	}
	
	long[][] WarshallFloyd(long[][] a){
		int n = a.length;
		long[][] ans = new long[n][n];
		for(int i=0;i<n;i++) {
			for(int j=0;j<n;j++) {
				ans[i][j] = a[i][j]==0?(long)1e16:a[i][j];
				if(i==j)ans[i][j]=0;
			}
		}
		for(int k=0;k<n;k++) {
			for(int i=0;i<n;i++) {
				for(int j=0;j<n;j++) {
					ans[i][j] = Math.min(ans[i][j],ans[i][k]+ans[k][j]);
				}
			}
		}
		return ans;
	}
	long[] maxtra(int s,long l){
		long[] L = new long[size];
		for(int i=0;i<size;i++){
			L[i] = -1;
		}
		int[] v = new int[size];
		L[s] = -1;

		PriorityQueue<Pair> Q = new PriorityQueue<Pair>(new SampleComparator());
		Q.add(new Pair(l,s));

		while(!Q.isEmpty()){

			Pair C = Q.poll();
			if(v[(int)C.b]==0){
				L[(int)C.b] = C.a;
				v[(int) C.b] = 1;
				for(Edge D:list[(int) C.b])Q.add(new Pair(Math.max(L[(int)C.b],D.cost),D.to));
			}
		}

		return L;
	}
	long[] mintra(int s){
		long[] L = new long[size];
		for(int i=0;i<size;i++){
			L[i] = -1;
		}
		int[] v = new int[size];
		L[s] = s;

		PriorityQueue<Pair> Q = new PriorityQueue<Pair>(new SampleComparator().reversed());
		Q.add(new Pair(s,s));

		while(!Q.isEmpty()){

			Pair C = Q.poll();
			if(v[(int)C.b]==0){
				L[(int)C.b] = C.a;
				v[(int) C.b] = 1;
				for(Edge D:list[(int) C.b])Q.add(new Pair(Math.min(L[(int)C.b],D.cost),D.to));
			}
		}

		return L;
	}
	long Kruskal(){
		long r = 0;
		for(int i=0;i<size;i++) {
			for(Edge e:list[i]) {
				Edges.add(new LinkEdge(e.cost,i,e.to));
			}
		}
		UnionFindTree UF = new UnionFindTree(size);
		for(LinkEdge e:Edges){
			if(e.a>=0&&e.b>=0) {
				if(!UF.same(e.a,e.b)){
					r += e.L;
					UF.unite(e.a,e.b);
				}
			}
		}
		return r;
	}

	ArrayList<Integer> Kahntsort(){

		ArrayList<Integer> ans = new ArrayList<Integer>();
		PriorityQueue<Integer> q = new PriorityQueue<Integer>();
		int[] in = new int[size];
		for(int i=0;i<size;i++) {
			for(Edge e:list[i])in[e.to]++;
		}
		for(int i=0;i<size;i++) {
			if(in[i]==0)q.add(i);
		}
		while(!q.isEmpty()) {
			int v = q.poll();
			ans.add(v);
			for(Edge e:list[v]) {
				in[e.to]--;
				if(in[e.to]==0)q.add(e.to);
			}
		}
		for(int i=0;i<size;i++) {
			if(in[i]>0)return new ArrayList<Integer>();
		}
		return ans;
	}

	RootedTree dfsTree(int i) {
		int[] u = new int[size];
		RootedTree r = new RootedTree(size);
		dfsTree(i,u,r);
		return r;

	}

	private void dfsTree(int i, int[] u, RootedTree r) {
		u[i] = 1;
		for(Edge e:list[i]) {
			if(u[e.to]==0) {
				r.list[i].add(e);
				u[e.to] = 1;
				dfsTree(e.to,u,r);
			}
		}

	}
}


class Tree extends Graph{

	public Tree(int N) {
		super(N);
	}

	long[] tyokkei(){
		long[][] a = bfs(0);

		System.out.println();
		int md = -1;
		long m = 0;
		for(int i=0;i<size;i++){
			if(m<a[0][i]){
				m = a[0][i];
				md = i;
			}
		}

		long[][] b = bfs(md);
		int md2 = -1;
		long m2 = 0;
		for(int i=0;i<size;i++){
			if(m2<b[0][i]){
				m2 = b[0][i];
				md2 = i;
			}
		}
		long[] r = {m2,md,md2};
		return r;
	}

}

class RootedTree extends Graph{
	RootedTree(int N){
		super(N);
	}
}
class LinkEdge{
	long L;
	int a ;
	int b;
	LinkEdge(long l,int A,int B){
		L = l;
		a = A;
		b = B;
	}
	public boolean equals(Object o){
		LinkEdge O = (LinkEdge) o;
		return O.a==this.a&&O.b==this.b&&O.L==this.L;
	}

	public int hashCode(){
		return Objects.hash(L,a,b);
	}
}

class Edge{
	int to;
	long cost;

	Edge(int a,long b){
		to = a;
		cost = b;
	}
}
class LinkEdgeComparator implements Comparator<LinkEdge>{
	public int compare(LinkEdge P, LinkEdge Q) {
		long t = P.L-Q.L;
		if(t==0){
			if(P.a>Q.a){
				return 1;
			}else{
				return P.b>Q.b?1:-1;
				
			}
		}
		return t>=0?1:-1;
	}
}

class Triplet{
	long a;
	long b;
	long c;
	Triplet(long p,long q,long r){
		a = p;
		b = q;
		c = r;
	}
	public boolean equals(Object o){
		Triplet O = (Triplet) o;
		return O.a==this.a&&O.b==this.b&&O.c==this.c?true:false;
			
	}

	public int hashCode(){
		return Objects.hash(a,b,c);
	}
}

class TripletComparator implements Comparator<Triplet>{
	public int compare(Triplet P, Triplet Q) {
		long t = P.a-Q.a;
		if(t==0){
			long tt = P.b-Q.b;
			if(tt==0) {
				if(P.c>Q.c) {
					return 1;
				}else if(P.c<Q.c){
					return -1;
				}else {
					return 0;
				}
			}
			return tt>0?1:-1;
		}
		return t>=0?1:-1;
	}
}
class Pair{
	long a;
	long b;

	Pair(long p,long q){
		this.a = p;
		this.b = q;
	}

	public boolean equals(Object o){
		Pair O = (Pair) o;
		return O.a==this.a&&O.b==this.b;
	}

	public int hashCode(){
		return Objects.hash(a,b);
	}
}

class SampleComparator implements Comparator<Pair>{
	public int compare(Pair P, Pair Q) {
		long t = P.a-Q.a;
		if(t==0){
			return P.b>=Q.b?1:-1;

		}
		return t>=0?1:-1;
	}
}


class LongIntPair{
	long a;
	int b;

	LongIntPair(long p,int q){
		this.a = p;
		this.b = q;
	}

	public boolean equals(Object o){
		Pair O = (Pair) o;
		return O.a==this.a&&O.b==this.b;

	}

	public int hashCode(){
		return Objects.hash(a,b);
	}
}

class LongIntSampleComparator implements Comparator<LongIntPair>{
	public int compare(LongIntPair P, LongIntPair Q) {
		long t = P.a-Q.a;
		if(t==0){
			if(P.b>Q.b){
				return 1;
			}else{
				return -1;
			}
		}
		return t>=0?1:-1;
	}
}


class IntIntPair{
	int a;
	int b;

	IntIntPair(int p,int q){
		this.a = p;
		this.b = q;
	}

	public boolean equals(Object o){
		Pair O = (Pair) o;
		return O.a==this.a&&O.b==this.b;
			
	}

	public int hashCode(){
		return Objects.hash(a,b);
	}
}



class IntIntSampleComparator implements Comparator<IntIntPair>{
	public int compare(IntIntPair P, IntIntPair Q) {
		int t = P.a-Q.a;
		if(t==0){
			return P.b-Q.b;
		}
		return t;
	}
}




class DoublePair{
	double a;
	double b;

	DoublePair(double p,double q){
		this.a = p;
		this.b = q;
	}

	public boolean equals(Object o){
		Pair O = (Pair) o;
		return O.a==this.a&&O.b==this.b;
			
	}

	public int hashCode(){
		return Objects.hash(a,b);
	}
}

class DDSampleComparator implements Comparator<DoublePair>{
	public int compare(DoublePair P, DoublePair Q) {
		return P.b-Q.b>=0?1:-1;
	}
}

class DoubleTriplet{
	double a;
	double b;
	double c;

	DoubleTriplet(double p,double q,double r){
		this.a = p;
		this.b = q;
		this.c = r;
	}

	public boolean equals(Object o){
		DoubleTriplet O = (DoubleTriplet) o;
		return O.a==this.a&&O.b==this.b&&O.c==this.c;
			
	}

	public int hashCode(){
		return Objects.hash(a,b,c);
	}
}


class FastScanner {
    private final java.io.InputStream in = System.in;
    private final byte[] b = new byte[1024];
    private int p = 0;
    private int bl = 0;
    private boolean hNB() {
        if (p<bl) {
            return true;
        }else{
            p = 0;
            try {
                bl = in.read(b);
            } catch (IOException e) {
                e.printStackTrace();
            }
            if (bl<=0) {
                return false;
            }
        }
        return true;
    }
    private int rB() { if (hNB()) return b[p++]; else return -1;}
    private static boolean iPC(int c) { return 33 <= c && c <= 126;}
    private void sU() { while(hNB() && !iPC(b[p])) p++;}
    public boolean hN() { sU(); return hNB();}
    public String next() {
        if (!hN()) throw new NoSuchElementException();
        StringBuilder sb = new StringBuilder();
        int b = rB();
        while(iPC(b)) {
            sb.appendCodePoint(b);
            b = rB();
        }
        return sb.toString();
    }
    public char nextChar() {
    	return next().charAt(0);
    }
    public long nextLong() {
        if (!hN()) throw new NoSuchElementException();
        long n = 0;
        boolean m = false;
        int b = rB();
        if (b=='-') {
            m=true;
            b=rB();
        }
        if (b<'0'||'9'<b) {
            throw new NumberFormatException();
        }
        while(true){
            if ('0'<=b&&b<='9') {
                n *= 10;
                n += b - '0';
            }else if(b == -1||!iPC(b)){
                return (m?-n:n);
            }else{
                throw new NumberFormatException();
            }
            b = rB();
        }
    }
    public int nextInt() {
        if (!hN()) throw new NoSuchElementException();
        long n = 0;
        boolean m = false;
        int b = rB();
        if (b == '-') {
            m = true;
            b = rB();
        }
        if (b<'0'||'9'<b) {
            throw new NumberFormatException();
        }
        while(true){
            if ('0'<=b&&b<='9') {
                n *= 10;
                n += b-'0';
            }else if(b==-1||!iPC(b)){
                return (int) (m?-n:n);
            }else{
                throw new NumberFormatException();
            }
            b = rB();
        }
    }
    public int[] nextInts(int n) {
    	int[] a = new int[n];
    	for(int i=0;i<n;i++) {
    		a[i] = nextInt();
    	}
    	return a;
    }
    public long[] nextLongs(int n) {
    	long[] a = new long[n];
    	for(int i=0;i<n;i++) {
    		a[i] = nextLong();
    	}
    	return a;
    }
    public int[][] nextIntses(int n,int m){
    	int[][] a = new int[n][m];
    	for(int i=0;i<n;i++) {
    		for(int j=0;j<m;j++) {
    			a[i][j] = nextInt();
    		}
    	}
    	return a;
    }
}


class Mathplus{
	int mod = 1000000007;
	long[] fac;
	long[] revfac;
	long[][] comb;
	long[] pow;
	long[] revpow;
	boolean isBuild = false;
	boolean isBuildc = false;
	boolean isBuildp = false;
	int mindex = -1;
	int maxdex = -1;
	
	int color(int[][] diff,int N) {
		
		int[] val = new int[1<<N];
		val[0] = 1;
		for(int i=0;i<(1<<N);i++) {		
			for(int j=0;j<N;j++) {
				if(contains(i,j)) {
					if(val[bitremove(i,j)]==1) {
						boolean b = true;
						for(int k=0;k<N;k++) {
							if(contains(i,k)&&diff[j][k]==1) {
								b = false;
							}
						}
						if(b)val[i] = 1;
					}
					break;
				}
			}
		}
		int[] dp = new int[1<<N];
		Arrays.fill(dp,mod);;
		dp[0] = 0;
		for(int i=0;i<(1<<N);i++) {
			for(int j=i;j>0;j=(j-1)&i) {
				if(val[j]==1)dp[i]=Math.min(dp[i],dp[i^j]+1);
			}
		}
		return dp[(1<<N)-1];
	}
	
	public DoubleTriplet Line(double x1,double y1,double x2,double y2) {
		double a = y1-y2;
		double b = x2-x1;
		double c = x1*y2-x2*y1;
		return new DoubleTriplet(a,b,c);
	}
	public double putx(DoubleTriplet T,double x) {
		return -(T.a*x+T.c)/T.b;
	}
	public double puty(DoubleTriplet T,double y) {
		return -(T.b*y+T.c)/T.a;
	}
	public double DistanceofPointandLine(DoublePair P,Triplet T) {
		return Math.abs(P.a*T.a+P.b*T.b+T.c) / Math.sqrt(T.a*T.a+T.b*T.b);
	}
	public boolean cross(long ax, long ay, long bx, long by, long cx, long cy, long dx, long dy) {
		 long ta = (cx - dx) * (ay - cy) + (cy - dy) * (cx - ax);
		 long tb = (cx - dx) * (by - cy) + (cy - dy) * (cx - bx);
		 long tc = (ax - bx) * (cy - ay) + (ay - by) * (ax - cx);
		 long td = (ax - bx) * (dy - ay) + (ay - by) * (ax - dx);
		 return((tc>=0&&td<=0)||(tc<=0&&td>=0))&&((ta>=0&&tb<=0)||(ta<=0&&tb>=0));
	}
	public boolean dcross(double ax, double ay, double bx, double by, double cx, double cy, double dx, double dy) {
		 double ta = (cx - dx) * (ay - cy) + (cy - dy) * (cx - ax);
		 double tb = (cx - dx) * (by - cy) + (cy - dy) * (cx - bx);
		 double tc = (ax - bx) * (cy - ay) + (ay - by) * (ax - cx);
		 double td = (ax - bx) * (dy - ay) + (ay - by) * (ax - dx);
		 return((tc>=0&&td<=0)||(tc<=0&&td>=0))&&((ta>=0&&tb<=0)||(ta<=0&&tb>=0));
	}
	
	void buildFac(){
		fac = new long[10000003];
		revfac = new long[10000003];
		fac[0] = 1;
		for(int i=1;i<=10000002;i++){
			fac[i] = (fac[i-1] * i)%mod;
		}
		revfac[10000002] = rev(fac[10000002])%mod;
		for(int i=10000001;i>=0;i--) {
			revfac[i] = (revfac[i+1] * (i+1))%mod;
		}
		isBuild = true;
	}
	public long[] buildrui(int[] a) {
		int n = a.length;
		long[] ans = new long[n];
		ans[0] = a[0];
		for(int i=1;i<n;i++) {
			ans[i] = ans[i-1] + a[i];
		}
		return ans;
	}
	public long[][] buildrui(int[][] a) {
		int n = a.length;
		int m = a[0].length;
		long[][] ans = new long[n][m];
		for(int i=1;i<n;i++) {
			for(int j=1;j<m;j++) {
				ans[i][j] = a[i][j];
			}
		}
		for(int i=1;i<n;i++) {
			for(int j=1;j<m;j++) {
				ans[i][j] += ans[i][j-1] + ans[i-1][j] - ans[i-1][j-1];
			}
		}
		return ans;
	}
	long divroundup(long n,long d) {
		if(n==0)return 0;
		return (n-1)/d+1;
	}
	public long sigma(long i) {
		return i*(i+1)/2;
	}
	public int digit(long i) {
		int ans = 1;
		while(i>=10) {
			i /= 10;
			ans++;
		}
		return ans;

	}
	public int popcount(int i) {
		int ans = 0;
		while(i>0) {
			ans += i%2;
			i /= 2;
		}
		return ans;
	}
	public boolean contains(int S,int i) {return (S>>i&1)==1;}
	public int bitremove(int S,int i) {return S&(~(1<<i));}
	public int bitadd(int S,int i) {return S|(1<<i);}
	public boolean isSubSet(int S,int T) {return (S-T)==(S^T);}
	public boolean isDisjoint(int S,int T) {return (S+T)==(S^T);}
	public int isBigger(int[] d, int i) {
		int ok = d.length;
		int ng = -1;
		while(Math.abs(ok-ng)>1) {
			int mid = (ok+ng)/2;
			if(d[mid]>i) {
				ok = mid;
			}else {
				ng = mid;
			}
		}
		return ok;
	}
	public int isSmaller(int[] d, int i) {
		int ok = -1;
		int ng = d.length;
		while(Math.abs(ok-ng)>1) {
			int mid = (ok+ng)/2;
			if(d[mid]<i) {
				ok = mid;
			}else {
				ng = mid;
			}
		}
		return ok;
	}
	public int isBigger(long[] d, long i) {
		int ok = d.length;
		int ng = -1;
		while(Math.abs(ok-ng)>1) {
			int mid = (ok+ng)/2;
			if(d[mid]>i) {
				ok = mid;
			}else {
				ng = mid;
			}
		}
		return ok;
	}
	public int isSmaller(long[] d, long i) {
		int ok = -1;
		int ng = d.length;
		while(Math.abs(ok-ng)>1) {
			int mid = (ok+ng)/2;
			if(d[mid]<i) {
				ok = mid;
			}else {
				ng = mid;
			}
		}
		return ok;
	}
	public int isBigger(ArrayList<Long> d, long i) {
		int ok = d.size();
		int ng = -1;
		while(Math.abs(ok-ng)>1) {
			int mid = (ok+ng)/2;
			if(d.get(mid)>i) {
				ok = mid;
			}else {
				ng = mid;
			}
		}
		return ok;
	}
	public int isSmaller(ArrayList<Long> d, long i) {
		int ok = -1;
		int ng = d.size();
		while(Math.abs(ok-ng)>1) {
			int mid = (ok+ng)/2;
			if(d.get(mid)<i) {
				ok = mid;
			}else {
				ng = mid;
			}
		}
		return ok;
	}
	public HashSet<Integer> primetable(int m) {
		HashSet<Integer> pt = new HashSet<Integer>();
		for(int i=2;i<=m;i++) {
			boolean b = true;
			for(int d:pt) {
				if(i%d==0) {
					b = false;
					break;
				}
			}
			if(b) {
				pt.add(i);
			}
		}
		return pt;
	}
	public ArrayList<Integer> primetablearray(int m) {
		ArrayList<Integer> al = new ArrayList<Integer>();
		Queue<Integer> q = new ArrayDeque<Integer>();
		for(int i=2;i<=m;i++) {
			q.add(i);
			
		}
		boolean[] b = new boolean[m+1];
		while(!q.isEmpty()) {
			int e = q.poll();
			if(!b[e]) {
				al.add(e);
				for(int j=1;e*j<=1000000;j++) {
					b[e*j] = true;
				}
			}
		}
		return al;
	}
	public HashMap<Integer,Integer> hipPush(ArrayList<Integer> l){
		HashMap<Integer,Integer> r = new HashMap<Integer,Integer>();
		TreeSet<Integer> s = new TreeSet<Integer>();
		for(int e:l)s.add(e);
		int p = 0;
		for(int e:s) {
			r.put(e,p);
			p++;
		}
		return r;
	}
	public TreeMap<Integer,Integer> thipPush(ArrayList<Integer> l){
		TreeMap<Integer,Integer> r = new TreeMap<Integer,Integer>();
		Collections.sort(l);
		int b = -(mod+9393);
		int p = 0;
		for(int e:l) {
			if(b!=e) {
				r.put(e,p);
				p++;
			}
			b=e;
		}
		return r;
	}

	long max(long[] a){
		long M = 0;
		for(int i=0;i<a.length;i++){
			if(M<a[i]){
				M =a[i];
				maxdex = i;
			}
		}
		return M;
	}
	int max(int[] a){
		int M = 0;
		for(int i=0;i<a.length;i++){
			if(M<a[i]){
				M =a[i];
				maxdex = i;
			}
		}
		return M;
	}
	long min(long[] a){
		long m = Long.MAX_VALUE;
		for(int i=0;i<a.length;i++){
			if(m>a[i]){
				m =a[i];
				mindex = i;
			}
		}
		return m;
	}
	int min(int[] a){
		int m = Integer.MAX_VALUE;
		for(int i=0;i<a.length;i++){
			if(m>a[i]){
				m =a[i];
				mindex = i;
			}
		}
		return m;
	}
	long sum(long[] a){
		long s = 0;
		for(int i=0;i<a.length;i++)s += a[i];
		return s;
	}
	long sum(int[] a){
		long s = 0;
		for(int i=0;i<a.length;i++)s += a[i];
		return s;
	}
	long gcd(long a, long b){
		a = Math.abs(a);
		b = Math.abs(b);
		if(a==0)return b;
		if(b==0)return a;
		if(a%b==0) return b;
		else return gcd(b,a%b);
	}
	int igcd(int a, int b) {
		if(a%b==0) return b;
		else return igcd(b,a%b);
	}
	long lcm(long a, long b) {return a / gcd(a,b) * b;}
	public long perm(int a,int num) {
		if(!isBuild)buildFac();
		return fac[a]*(rev(fac[a-num]))%mod;
	}
	void buildComb(int N) {
		comb = new long[N+1][N+1];
		comb[0][0] = 1;
		for(int i=1;i<=N;i++) {
			comb[i][0] = 1;
			for(int j=1;j<N;j++) {
				comb[i][j] = comb[i-1][j-1]+comb[i-1][j];
				if(comb[i][j]>mod)comb[i][j]-=mod;
			}
			comb[i][i] = 1;
		}
	}
	public long comb(int a,int num){
		if(a-num<0)return 0;
		if(num<0)return 0;
		if(!isBuild)buildFac();
		return fac[a] * ((revfac[num]*revfac[a-num])%mod)%mod;
	}
	long mulchoose(int n,int k) {
		if(k==0) return 1;
		return comb(n+k-1,k);
	}
	long rev(long l) {return pow(l,mod-2);}
	
	void buildpow(int l,int i) {
		pow = new long[i+1];
		pow[0] = 1;
		for(int j=1;j<=i;j++) {
			pow[j] = pow[j-1]*l;
			if(pow[j]>mod)pow[j] %= mod;
		}
	}
	void buildrevpow(int l,int i) {
		revpow = new long[i+1];
		revpow[0] = 1;
		for(int j=1;j<=i;j++) {
			revpow[j] = revpow[j-1]*l;
			if(revpow[j]>mod) revpow[j] %= mod;		
		}
	}
	long pow(long l, long i) {
		if(i==0)return 1;
		else{
			if(i%2==0){
				long val = pow(l,i/2);
				return val * val % mod;
			}
			else return pow(l,i-1) * l % mod;
		}
	}
}