Problem Statement

For a set $X$ of non-negative integers, let $f(X)$ denote the set of non-negative integers that can be represented as the bitwise $\\mathrm{XOR}$ of two integers (possibly the same) in $X$. As an example, for $X=\\lbrace 1, 2, 4\\rbrace$, we have $f(X)=\\lbrace 0, 3, 5, 6\\rbrace$.

You are given a set of $N$ non-negative integers less than $2^M$: $S=\\lbrace A_1, A_2, \\dots, A_N\\rbrace$.

You may perform the following operation any number of times.

• Divide $S$ into two sets $T_1$ and $T_2$ (one of them may be empty). Then, replace $S$ with the union of $f(T_1)$ and $f(T_2)$.

Find the minimum number of operations needed to make $S=\\lbrace 0\\rbrace$.

What is bitwise $\\mathrm{XOR}$?

The bitwise $\\mathrm{XOR}$ of non-negative integers $A$ and $B$, $A \\oplus B$, is defined as follows.

• When $A \\oplus B$ is written in binary, the $k$-th lowest bit ($k \\geq 0$) is $1$ if exactly one of the $k$-th lowest bits of $A$ and $B$ is $1$, and $0$ otherwise.

For instance, $3 \\oplus 5 = 6$ (in binary: $011 \\oplus 101 = 110$).
Generally, the bitwise $\\mathrm{XOR}$ of $k$ non-negative integers $p_1, p_2, p_3, \\dots, p_k$ is defined as $(\\dots ((p_1 \\oplus p_2) \\oplus p_3) \\oplus \\dots \\oplus p_k)$, which can be proved to be independent of the order of $p_1, p_2, p_3, \\dots, p_k$.

Constraints

• $1 \\leq N \\leq 300$
• $1 \\leq M \\leq 300$
• $0 \\leq A_i < 2^{M}$
• $A_i\\ (1\\leq i \\leq N)$ are pairwise distinct.
• Each $A_i$ is given with exactly $M$ digits in binary (possibly with leading zeros).
• All values in the input are integers.

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Input

The input is given from Standard Input in the following format:

$N$ $M$ $A_1$ $A_2$ $\\vdots$ $A_N$

Output

Print the answer.

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Sample Input 1

4 2
00
01
10
11

Sample Output 1

2

In the first operation, you can divide $S$ into $T_1=\\lbrace 0, 1\\rbrace, T_2=\\lbrace 2, 3\\rbrace$, for which $f(T_1) =\\lbrace 0, 1\\rbrace, f(T_2) =\\lbrace 0, 1\\rbrace$, replacing $S$ with $\\lbrace 0, 1\\rbrace$.

In the second operation, you can divide $S$ into $T_1=\\lbrace 0\\rbrace, T_2=\\lbrace 1\\rbrace$, making $S=\\lbrace 0\\rbrace$.

There is no way to make $S=\\lbrace 0\\rbrace$ in fewer than two operations, so the answer is $2$.

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Sample Input 2

1 8
10011011

Sample Output 2

1

In the first operation, you can divide $S$ into $T_1=\\lbrace 155\\rbrace, T_2=\\lbrace \\rbrace$, making $S=\\lbrace 0\\rbrace$. Note that $T_1$ or $T_2$ may be empty.

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Sample Input 3

1 2
00

Sample Output 3

0

We have $S=\\lbrace 0\\rbrace$ from the beginning; no operation is needed.

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Sample Input 4

20 20
10011011111101101111
10100111100001111100
10100111100110001111
10011011100011011111
11001000001110011010
10100111111011000101
11110100001001110010
10011011100010111001
11110100001000011010
01010011101011010011
11110100010000111100
01010011101101101111
10011011100010110111
01101111101110001110
00111100000101111010
01010011101111010100
10011011100010110100
01010011110010010011
10100111111111000001
10100111111000010101

Sample Output 4

10