Problem: You are given a set S of strings consisting of `0` and `1`, and an integer K.

Find the longest string that is a subsequence of K or more different strings in S. If there are multiple strings that satisfy this condition, find the lexicographically smallest such string.

Here, S is given in the format below:

* The data directly given to you is an integer N, and N+1 strings X_0,X_1,...,X_N. For every i (0\leq i\leq N), the length of X_i is 2^i.
* For every pair of two integers (i,j) (0\leq i\leq N,0\leq j\leq 2^i-1), the j-th character of X_i is `1` if and only if the binary representation of j with i digits (possibly with leading zeros) belongs to S. Here, the first and last characters in X_i are called the 0-th and (2^i-1)-th characters, respectively.
* S does not contain a string with length N+1 or more.



Here, a string A is a subsequence of another string B when there exists a sequence of integers t_1 < ... < t_{|A|} such that, for every i (1\leq i\leq |A|), the i-th character of A and the t_i-th character of B is equal.

Constraints

* 0 \leq N \leq 20
* X_i(0\leq i\leq N) is a string of length 2^i consisting of `0` and `1`.
* 1 \leq K \leq |S|
* K is an integer.

Input

Input is given from Standard Input in the following format:


N K
X_0
:
X_N


Output

Print the lexicographically smallest string among the longest strings that are subsequences of K or more different strings in S.

Examples

Input

3 4
1
01
1011
01001110


Output

10


Input

4 6
1
01
1011
10111010
1101110011111101


Output

100


Input

2 5
0
11
1111


Output

Solution (in CPP):

Solution:
#include<bits/stdc++.h>
#define ll long long
#define maxn 4097152
using namespace std;
int N,k;
int ha[maxn][21],res[maxn][21],dc[maxn];
int Gstat(int s1,int s2,int l1,int l2)
{
	assert((((1<<l1)+s1)<<l2)+s2<maxn);
	return (((1<<l1)+s1)<<l2)+s2;
}
char buf[21][maxn];
int ans,alen;
void Pre()
{
	int inv,tinv,ginv;
	for(int i=0;i<=N;i++) scanf("%s",buf[i]);
	alen=1e9;
	for(int i=N;i>=0;i--)
	{
		//printf("%d\n",i);
		for(int j=0;j<=i;j++)
		{
			for(int fr=0;fr<(1<<j);fr++)
			{
				for(int tl=0;tl<(1<<(i-j));tl++)
				{
					//assert(inc<maxn*60);
					int stat=Gstat(fr,tl,j,i-j);
					if(j==0) ha[stat][j]=(buf[i][tl]=='1');
					
					if(tl!=0) ha[Gstat(fr<<1|1,tl^(1<<dc[tl]-1),j+1,dc[tl]-1)][j+1]+=ha[stat][j];
					
					if(tl!=(1<<(i-j))-1)
					{
						inv=(tl^((1<<(i-j))-1));
						tinv=(inv^(1<<dc[inv]-1));
						ginv=(tinv^((1<<(dc[inv]-1))-1));
						ha[Gstat(fr<<1,ginv,j+1,dc[inv]-1)][j+1]+=ha[stat][j];
					}
					res[fr][j]+=ha[stat][j];
				}
			}
		}
	}
	for(int i=N;i>=0;i--)
		for(int j=0;j<(1<<i);j++)
			if(res[j][i]>=k)
			{
				for(int t=i-1;t>=0;t--) printf("%d",j>>t&1);
				return;
			}
}
int main()
{
	scanf("%d%d",&N,&k);
	for(int i=1;i<maxn;i++) dc[i]=dc[i>>1]+1;
	dc[0]=1;
	Pre();
	return 0;
}