A game starts with a Uniformly random integer . 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 or divide by . If the number is odd, the player can either subtract or subtract , then subsequently divide by . The game ends when the number reaches , and the player who reduces from to wins. Alice gets the first move in the game. Assume that Alice and Bob both play optimally. Let be the probability that Bob wins when the upper bound is . If it exists, find .