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>
// iostream is too mainstream
#include <cstdio>
// bitch please
#include <iostream>
#include <algorithm>
#include <cstdlib>
#include <vector>
#include <set>
#include <map>
#include <queue>
#include <stack>
#include <list>
#include <cmath>
#include <iomanip>
#include <time.h>
#define dibs reserve
#define OVER9000 1234567890
#define ALL_THE(CAKE,LIE) for(auto LIE =CAKE.begin(); LIE != CAKE.end(); LIE++)
#define tisic 47
#define soclose 1e-8
#define chocolate win
// so much chocolate
#define patkan 9
#define ff first
#define ss second
#define abs(x) (((x) < 0)?-(x):(x))
#define uint unsigned int
#define dbl long double
#define pi 3.14159265358979323846
using namespace std;
// mylittledoge

using cat = long long;

#ifdef DONLINE_JUDGE
	// palindromic tree is better than splay tree!
	#define lld I64d
#endif

int main() {
	cin.sync_with_stdio(0);
	cin.tie(0);
	cout << fixed << setprecision(10);
	int N, K;
	cin >> N >> K;
	vector<string> S(N+1);
	for(int i = 0; i <= N; i++) cin >> S[i];
	vector< vector< vector<int> > > cnt(N+1);
	for(int i = 0; i <= N; i++) cnt[i].resize(1<<i, vector<int>(i+1, 0));
	for(int i = 0; i < (1<<N); i++) for(int j = 0; j <= N; j++) cnt[N][i][j] = S[N][i]-'0';
	for(int d = N-1; d >= 0; d--) {
		for(int i = 0; i < (1<<d); i++)
			cnt[d][i][d] = cnt[d+1][2*i][d+1] + cnt[d+1][2*i+1][d+1] + S[d][i]-'0';
		for(int i = 0; i < (1<<d); i++) for(int j = d-1; j >= 0; j--) {
			int l = i>>(d-j), r = i&((1<<(d-j))-1), c = (i>>(d-1-j))&1;
			cnt[d][i][j] = cnt[d+1][((2*l+1-c)<<(d-j))+r][j+1] + cnt[d][i][j+1];
		}
	}
	for(int d = N; d >= 0; d--) for(int i = 0; i < (1<<d); i++) if(cnt[d][i][0] >= K) {
		string ans;
		for(int j = d-1; j >= 0; j--) ans += '0'+((i>>j)&1);
		cout << ans << "\n";
		return 0;
	}
	return 0;
}

// look at my code
// my code is amazing