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 <algorithm>

using namespace std;

namespace io
{
	const int maxb = 1 << 15;
	char b[maxb], *s = b , *t = b;
	bool Getchar(char &ch)
	{
		return ch = s == t && (t = (s = b) + fread(b, 1, maxb, stdin)) == b ? 0 : *s ++;
	}
	int Getint()
	{
		char ch;
		while (Getchar(ch) && (ch < '0' || ch > '9'));
		int s = ch - '0';
		while (Getchar(ch) && ch >= '0' && ch <= '9')
			s = s * 10 + ch - '0';
		return s;
	}
}

using io::Getchar;
using io::Getint;

int f[21][1 << 20 | 1];

int main()
{
	int n = Getint(), K = Getint();
	
	for (int i = 0; i <= n; ++ i)
	{
		int all = 1 << i;
		char ch;
		while (Getchar(ch) && ch != '0' && ch != '1');
		for (int j = 0; j < all; ++ j)
		{
			f[i][j] = ch - '0';
			Getchar(ch);
		}
	}
	
	for (int pos = 0; pos < n; ++ pos)
	{
		for (int i = n; i > pos; -- i)
		{
			int all = 1 << i;
			for (int j = 0; j < all; ++ j)
			{
				if (!f[i][j]) continue;
				if (pos && (j >> pos & 1) == (j >> pos - 1 & 1)) continue;
				f[i - 1][(j >> pos + 1) << pos | (j & (1 << pos) - 1)] += f[i][j];
			}
		}
	}
	
	for (int i = n; ~ i; -- i)
	{
		int all = 1 << i;
		for (int j = 0; j < all; ++ j)
		{
			if (f[i][j] >= K)
			{
				for (int pos = i - 1; ~ pos; -- pos)
					putchar((j >> pos & 1) + '0');
				return 0;
			}
		}
	}
}