There are balls in an urn. Alice and Bob pick the balls in turn, for each round they can pick balls for any choice of 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 be the position of Alice ( if she should go first, if she should go second). Let be the maximum number of balls Alice should select on her first turn if Bob chooses balls on his first turn. Find .