Game on a Strip

Tzuyu gave Nayeon a strip of $N$ cells (numbered $1$ through $N$) for her birthday. This strip is described by a sequence $A_1, A_2, \ldots, A_N$, where for each valid $i$, the $i$-th cell is blocked if $A_i = 1$ or free if $A_i = 0$. Tzuyu and Nayeon are going to use it to play a game with the following rules: - The players alternate turns; Nayeon plays first. - Initially, both players are ou