Take Not Less

Zenyk and Marichka are playing a game taking turns alternatively. They have $N$ piles of stones. In the $i^{th}$ pile, there are $A_i$ stones initially. Marichka goes first, and removes any **non-zero** number of stones from any pile. After that, in one move, a player can take any number of stones from any pile, **but not less** than the number of stones taken by the opponent in the previous m