Black&White Ring Game

Alice and Bob are playing a game. Alice goes first. They have a binary circular array $A$ with $N$ elements. In one move, a player can: - Choose a *circular continuous segment* of the array and perform a *left cyclic shift* on the chosen segment. We define the term $diff(A)$ as the number of pairs of **adjacent** elements with **different** values. Note that we also consider $A_N$ and $A_1$ as a