Partial Sums

You have a matrix $A\_0$, consisting of $n$ rows and $m$ columns. Rows and columns are numbered with consecutive natural numbers starting from $1$. The elements of the matrix are zeros and ones. Denote the element of this matrix at the intersection of the $i$ row and $j$ column as $A\_0\[i, j]$.

Consider an infinite sequence of matrices $A\_k$. The matrix $A\_k$ ($k > 0$) also consists of $n$ rows and $m$ columns and it is a matrix of partial sums for the matrix $A\_{k - 1}$ modulo $2$. Formally, this means that $A\_k\[i, j] = \sum\_{1 \le u \le i} \sum\_{1 \le v \le j} A\_{k - 1}\[u, v] \mod 2$

It is required to find a minimum $k > 0$ such that the matrices $A\_k$ and $A\_0$ are element-wise equal.

The first line of the input data contains two integers $n$ and $m$ --- the number of rows and columns in the matrix $A\_0$. The following $n$ lines contain descriptions of the rows of the matrix. Each line consists of $m$ characters, each character is either $0$ or $1$.

Output the single number $k$ --- the answer to the problem.

• $1 \le n, m \le 10^6$
• $n \times m \le 10^6$