Adjacent Flips

You are given a *binary* string $S$ of length $N$. You can perform the following operation on it: - Choose an index $i$ ($1 \leq i \lt N$) such that $S_i = S_{i+1}$, and flip both $S_i$ and $S_{i+1}$. Flipping $S_i$ means it becomes $1$ if it was originally $0$, and vice versa. For example, if $S = \texttt{0101100}$, performing the operation with $i = 4$ would result in $S = \texttt{010\underl