Greedy Grid Moves

Anya and Becky found an $N\times M$ grid that contains **distinct** integers, and decided to play a game on it. The integer present at the intersection of the $i$-th row and $j$-th column is denoted $A_{i, j}$. They place a token at the top-left corner (cell $(1, 1)$), and then take turns moving the token towards the bottom-right corner (cell $(N, M)$), each time moving the token one step right o