Shift Game

Alice is given two binary strings $A$ and $B$, each of length $N$. A binary string is a string consisting only of the characters `0` and `1`. Alice plays the following game: Initially, her score is $0$. In each iteration: - She adds the product of the first characters of the current strings to her score. That is, she adds $A_1 \cdot B_1$ to the score. - If $A_1 = B_1$, she replaces $A$ with