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:
/* ***********************************************
Author        :dasinlsb
Created Time  :2018/6/2 21:38:23
File Name     :C:\dasin\duipai.cpp
************************************************ */

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
#define pb push_back
#define fi first
#define se second
const int N=21;
const int M=(1<<20)+5;
int n,kk,f[N][M],nxt[N][M];
char Xs[N];
int main(){
  //freopen("C:/dasin/aa.in","r",stdin);
  //freopen("C:/dasin/mine.out","w",stdout);
  int i,j,k;
  scanf("%d%d",&n,&kk);
  for(i=0;i<=n;++i){
    scanf("%s",Xs);
    for(j=0;j<1<<i;++j)if(Xs[j]-'0')f[i][j]=1;
  }
  for(i=1;i<=n;++i){
    for(j=0;j<1<<i;++j){
      int t=j>>(i-1);
      for(k=i-1;~k&&((j>>k)&1)==t;--k);
      nxt[i][j]=k+1;
    }
  }
  int x=-1,y=-1;
  for(i=0;i<=n;++i){
    for(j=1;j+i<=n;++j){
      for(k=0;k<1<<i+j;++k){
        f[i][k>>j]+=f[i+j][k];
        int t=nxt[j][((1<<j)-1)&k];
        if(!t)continue;
        f[i+t][k>>j<<t|((1<<t)-1)&k]+=f[i+j][k];
      }
    }
    for(j=(1<<i)-1;~j;--j){
      if(f[i][j]>=kk){
        x=i,y=j;
      }
    }
  }
  assert(~x);
  for(i=x-1;~i;--i)putchar(((y>>i)&1)+'0');puts("");
  return 0;
}