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<cstdio>
#include<iostream>
using namespace std;
int n,k,p;
int bit[25];
int f[1<<21][21];
char X[(1<<20)+5];
int main()
{
	scanf("%d%d",&n,&k);
	bit[0]=1;
	for(int i=1;i<=n;++i)
	{
		bit[i]=bit[i-1]<<1;
	}
	for(int len=0;len<=n;++len)
	{
		scanf("%s",X);
		for(int s=0;s<bit[len];++s) if(X[s]=='1')
		{
			f[bit[len]|s][len]=1;
		}
	}
	for(int len=n;len>=0;--len)
	{
		for(int s=0;s<bit[len];++s)
		{
			for(int i=len;i>0;--i) if(f[bit[len]|s][i])
			{
				int a=s>>i;
				for(int j=i;j>0;--j)
				{
					if(s&bit[j-1])
					{
						f[1<<(len-i+j)|(a<<j|(s&(bit[j]-1)))][j-1]+=f[bit[len]|s][i];
						break;
					}
				} 
				for(int j=i;j>0;--j)
				{
					if(~s&bit[j-1])
					{
						f[1<<(len-i+j)|(a<<j|(s&(bit[j]-1)))][j-1]+=f[bit[len]|s][i];
						break;
					}
				}
				f[1<<(len-i)|a][0]+=f[bit[len]|s][i];
			}
			if(f[bit[len]|s][0]>=k)
			{
				for(int i=len-1;~i;--i)
				{
					printf("%d",bool(s&bit[i]));
				}
				printf("\n");
				return 0;
			}
		}
	}
	return 0;
}