Codeforces1184A2-Heidi Learns Hashing (Medium)

Here are the official tutorial：
A2. Medium
Let us first find a method to check whether a shift by a fixed number $k$ yields a solution or not. For the sake of simplicity let us work with the case of $n=12$, let also $k=3$. We can start by writing a set of equations
(all modulo $2$)
$(1)$
$$\{\begin{array}{l}{x}_0+x_3=y_0\\ x_3+x_6=y_3\\ x_6+x_9=y_6\\ x_9+x_0=\end{array}$$