Short Piles

Alice and Bob play a turn‐based game using several piles of stones. There are three types of piles: - $X$ piles each containing $1$ stone, - $Y$ piles each containing $2$ stones, and - $Z$ piles each containing $3$ stones. The players take turns with **Alice moving first**. On each turn, the current player must remove exactly **one stone** from any non-empty pile. **Alice is not allowed to remo