Shall we play a game

Alice and Bob play a game over $N$ rounds, starting with an initial score of $0$. Alice takes the first turn. In the $i_{th}$ round ($1 \leq i \leq N$), the current player must either add or subtract $2^{i-1}$ from the current score. As the game progresses, let: - $X$ be the maximum score reached at any point. - $Y$ be the minimum score reached at any point. Alice’s goal is to minimize the fin