Exponential Ball Draw

There are $100$ balls in an urn. Alice and Bob pick the balls in turn, for each round they can pick $2^k$ balls for any choice of $k \geq 0$ as long as there are at least that many balls remaining. Whoever draws the last remaining ball in the urn loses. Alice can choose whether to go first or second. Let $p$ be the position of Alice ($p = 1$ if she should go first, $p = 2$ if she should go second). Let $b$ be the maximum number of balls Alice should select on her first turn if Bob chooses $32$ balls on his first turn. Find $100p + b$.