Lucky Number Game

Bob and Alice are playing a game with the following rules: - Initially, they have an integer sequence $A_1, A_2, \ldots, A_N$; in addition, Bob has a *lucky number* $a$ and Alice has a lucky number $b$. - The players alternate turns. In each turn, the current player must remove a non-zero number of elements from the sequence; each removed element should be a multiple of the lucky number of this pl