Stingy Arrays

An array $B$ is said to be *stingy* if no two of its adjacent elements are equal. That is, $B$ is *stingy* iff $B_i \neq B_{i+1}$ for each $1 \leq i \lt |B|$. Alice and Bob have an array $A$ with them, and decided to play a game on it. They alternate turns, with Alice going first. On their turn, a player chooses any [subsequence](https://en.wikipedia.org/wiki/Subsequence) $S$ of $A$, such th