Reduction Mania

A game starts with a Uniformly random integer $N \sim \text{DiscreteUnif}\{1,2,\dots, n\}$. Alice and Bob take turns modifying the integer. A legal move depends on whether the integer is odd or even. If it's even, the player can either subtract $1$ or divide by $2$. If the number is odd, the player can either subtract $1$ or subtract $1$, then subsequently divide by $2$. The game ends when the number reaches $0$, and the player who reduces from $1$ to $0$ wins. Alice gets the first move in the game. Assume that Alice and Bob both play optimally. Let $p(n)$ be the probability that Bob wins when the upper bound is $n$. If it exists, find $\displaystyle \lim_{n \rightarrow \infty} p(n)$.