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>
using namespace std;

namespace TYC
{
	const int N = 20;
	
	int *F[N + 1][N + 1], n, k;
	char str[1 << N | 3];
	
	void work()
	{
		scanf("%d%d", &n, &k);
		for (int i = 0; i <= n; i++)
			for (int j = 0; i + j <= n; j++)
			{
				int len = 1 << (i + j) | 3;
				F[i][j] = new int[len];
				memset(F[i][j], 0, sizeof(int[len]));
			}
		for (int i = 0; i <= n; i++)
		{
			scanf("%s", str);
			for (int j = 0; j < 1 << i; j++)
				if (str[j] == '1')
					F[0][i][j] = 1;
		}
		for (int i = 0, v; i <= n; i++)
			for (int j = 1; i + j <= n; j++)
				for (int s = 0; s < 1 << (i + j); s++)
					if ((v = F[i][j][s]))
					{
						int d = (s >> (j - 1)) & 1;
						F[i + 1][j - 1][s] += v;
						F[i][0][s >> j] += v;
						for (int l = j - 1; l > 0; l--)
							if (((s >> (l - 1)) & 1) != d)
							{
								int t = ((s >> j) << l) | (s & ((1 << l) - 1));
								F[i + 1][l - 1][t] += v;
								break;
							}
					}
		for (int i = n; ~i; i--)
			for (int j = 0; j < 1 << i; j++)
				if (F[i][0][j] >= k)
				{
					for (int l = i - 1; ~l; l--)	
						putchar('0' + ((j >> l) & 1));
					return;
				}
	}
}

int main()
{
	TYC::work();
	return 0;
}