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:
R=range;N,K=map(int,input().split());Q=R(N+1);d=[[0]*2**i for i in Q];f=[[0]*2**i for i in Q]
for i in Q:
 for j,c in enumerate(input()):d[i][j]=int(c)
for i in R(1,N+1):
 for j in R(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 Q:
 for k in R(i+1,N+1):
  z=k-i;m=(1<<z)-1
  for j in R(1<<k):
   d[i][j>>z]+=d[k][j];r=f[z][j&m]
   if r!=z:d[k-r][j>>z<<z-r|j&(1<<z-r)-1]+=d[k][j]
 for j in R(1<<i):
  if d[i][j]>=K:I=i;J=j;break
print(''if I==J==0else bin(J)[2:].zfill(I))