Assistance Required

Dr. Goodwill is in a pickle! The evil Prof. Hades has kidnapped and reprogrammed his beloved robots Alice (called $A$ from here on) and Bob (called $B$ from here on) and made them the part of a cruel game. He has placed $A$ and $B$ on a $(2N+1) \times M$ board (coordinate system used for the board is $(X,Y)$ where $1 \le X \le (2N+1), 1\le Y \le M$). The board is made up of black and white tiles.