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 PYTHON3):

Solution:
N, K = map(int, input().split())
D = N + 1

d = [[0] * (1 << N) for _ in range(D)]
f = [[0] * (1 << N) for _ in range(D)]

for i in range(D):
    for j, c in enumerate(input()):
        if c == '1':
            d[i][j] = 1

for i in range(1, D):
    for j in range(1 << i):
        t = (j >> (i - 1)) & 1
        r = 0
        while r < i and ((j >> (i - 1 - r)) & 1) == t:
            r += 1
        f[i][j] = r

for i in range(D):
    for ii in range(i + 1, D):
        z = ii - i
        mask = (1 << z) - 1
        for j in range(1 << ii):
            d[i][j >> z] += d[ii][j]
            r = f[z][j & mask]
            if r != z:
                d[ii - r][((j >> z) << (z - r)) | (j & ((1 << (z - r)) - 1))] += d[ii][j]
    for j in range(1 << i):
        if d[i][j] >= K:
            I = i
            J = j
            break

print('' if I == J == 0 else bin(J)[2:].zfill(I))