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>
#define M 2197152
#define N 22
int d[M],e[M],f[M][N],g[M],h[M],i,j,m,n;char c[M];
int main()
{
    for(scanf("%d%d",&n,&m),i=2,j=1<<n+1;i<j;i++)d[i]=i&1?d[i>>1]+1:0,e[i]=i&1?0:e[i>>1]+1,h[i]=h[i>>1]+1;
    for(i=1;i<=1<<n;i<<=1)for(scanf("%s",c),e[i]=n,j=0;j<i;j++)f[i|j][0]=c[j]^48;
    for(i=1<<n+1;--i;)for(j=0;j<=h[i];j++)f[i&(1<<j)-1|i>>d[i>>j]&-(1<<j)][j+1]+=f[i][j],f[i&(1<<j)-1|i>>e[i>>j]&-(1<<j)][j+1]+=f[i][j],g[i&(1<<j)-1|1<<j]+=f[i][j];
    for(i=1,n=1<<n+1,j=0;i<n;i++)if(m<=g[i]&&h[j]<h[i])j=i;
    for(i=h[j];i--;i)putchar(j>>i&1^48);
    return 0*puts("");
}