String Game

Alice and Bob are playing a game on a binary string $S$ of length $N$. Alice and Bob alternate turns, with Alice going first. On their turn, the current player must choose any index $1 \le i \lt |S|$ such that $S_i \neq S_{i+1}$ and erase either $S_i$ or $S_{i+1}$ from the string. This reduces the length of $S$ by one, and the remaining parts of $S$ are concatenated in the same order. The player