Problem Statement

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 $S$.