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:
// includes {{{
#include<iostream>
#include<iomanip>
#include<algorithm>
#include<vector>
#include<stack>
#include<queue>
#include<deque>
#include<map>
#include<set>
#include<tuple>
#include<cmath>
#include<random>
#include<cassert>
// }}}
// #include<bits/stdc++.h>
using namespace std;
using ll = long long;

#include<bitset>

int n, k;
char buf[(1<<20)+10];
int dp[1<<21][21];

int main() {
  std::ios::sync_with_stdio(false), std::cin.tie(0);
  cin >> n >> k;
  for(int i = 0; i <= n; i++) {
    cin >> buf;
    for(int j = 0; j < (1 << i); j++) {
      if(buf[j] == '1') {
        int x = (1 << i) | j;
        dp[x][0] = 1;
      }
    }
  }

  for(int L = n; L > 0; --L) {
    for(int i = 0; i < (1 << L); i++) {
      int x = (1 << L) | i;
      int next0[21], next1[21];
      // dump(bitset<8>());
      // dump(bitset<8>(x));
      next0[L] = next1[L] = L;
      for(int h = L - 1; h >= 0; h--) {
        int f = (i >> (L - h - 1)) & 1;
        next0[h] = f == 0 ? h : next0[h+1];
        next1[h] = f == 1 ? h : next1[h+1];
      }
      for(int h = 0; h < L; h++) {
        int nx = x >> (L - h);
        // dump(h);
        // dump(dp[x][h]);
        // dump(bitset<8>(nx));
        // dump(h, next0[h+1], next1[h+1]);
        dp[nx][h] += dp[x][h];
        if(next0[h] < L) {
          int j0 = (i >> (L - h) << (L - next0[h])) | (i & ((1 << (L - next0[h])) - 1));
          int x0 = (1 << (L - next0[h] + h)) | j0;
          dp[x0][h+1] += dp[x][h];
        }
        if(next1[h] < L) {
          int j0 = (i >> (L - h) << (L - next1[h])) | (i & ((1 << (L - next1[h])) - 1));
          int x0 = (1 << (L - next1[h] + h)) | j0;
          dp[x0][h+1] += dp[x][h];
        }
      }
    }
  }
  for(int L = n; L >= 0; --L) {
    for(int i = 0; i < (1 << L); i++) {
      int x = (1 << L) | i;
      if(dp[x][L] >= k) {
        for(int h = L - 1; h >= 0; h--) {
          cout << ((i >> h) & 1);
        }
        cout << endl;
        return 0;
      }
    }
  }
  return 0;
}