Game of Pooks

We have found a new chess character — pook. It has the qualities of both a rook and a pawn. Specifically, treating the chessboard to be an $N\times N$ grid where $(i, j)$ denotes the intersection of the $i$-th row and the $j$-th column, a pook placed at square $(x, y)$ threatens the following squares: - $(i, y)$ for every $1 \leq i \leq N$ - $(x, i)$ for every $1 \leq i \leq N$ - $(x+1, y-1)$, if