Binary Strings Yet Again

For a ($0$-indexed) binary string $S$ of length $N$, define $f(S) = $ the number of pairs $(i, j) $ such that $0 \leq i \lt j \lt N$ and $S_i = 0, S_j = 1$. Given a binary string $S$ of size $N$, you can perform the following operation **exactly once**: - Choose an integer $K$ such that $1 \lt K \le N$ and $N \bmod K = 0$, and divide the string into continuous substrings of size $K$, i.e. you wil