There is given an integer $k$ and a grid $2^k \times 2^k$ with some numbers written in its cells, cell $(i, j)$ initially contains number $a_{ij}$. Grid is considered to be a torus, that is, the cell to the right of $(i, 2^k)$ is $(i, 1)$, the cell below the $(2^k, i)$ is $(1, i)$ There is also given a lattice figure $F$, consisting of $t$ cells, where $t$ is odd. $F$ doesn't have to be connected.

We can perform the following operation: place $F$ at some position on the grid. (Only translations are allowed, rotations and reflections are prohibited). Now choose any nonnegative integer $p$. After that, for each cell $(i, j)$, covered by $F$, replace $a_{ij}$ by $a_{ij}\oplus p$, where $\oplus$ denotes the bitwise XOR operation.

More formally, let $F$ be given by cells $(x_1, y_1), (x_2, y_2), \dots, (x_t, y_t)$. Then you can do the following operation: choose any $x, y$ with $1\le x, y \le 2^k$, any nonnegative integer $p$, and for every $i$ from $1$ to $n$ replace number in the cell $(((x + x_i - 1)\bmod 2^k) + 1, ((y + y_i - 1)\bmod 2^k) + 1)$ with $a_{((x + x_i - 1)\bmod 2^k) + 1, ((y + y_i - 1)\bmod 2^k) + 1}\oplus p$.

Our goal is to make all the numbers equal to $0$. Can we achieve it? If we can, find the smallest number of operations in which it is possible to do this.