Alice and Bob Play a Game

Alice challenges Bob to a game of Nimtraction. The game has the following rules: 1. Game starts with $n$ piles of matches with $a_1, a_2, \ldots ,a_n$ matches in each and a number $k$. 2. In each turn a player can choose to either remove any number of matches from one of the piles or subtract at most $z$ from the number $k$. 3. The game continues with the modified setup. It ends when $k$ becomes $