Alice and Bob start a game with a stack of $n$ cards. Let $n$ be chosen by rolling a fair $1000$ sided die. On their turn, each player will split the current stack of cards into two piles and discard the smaller pile. If both piles are the same size, one pile is still discarded. If a player is unable to split the stack of cards on their turn, they lose. If Alice goes first and both players play optimally, what is the probability that she wins?