Median Updates

For a binary string $S$ of length $N$, we define $f(S)$ as another string of length $N$, where the $i$-th character $f(S)_i$ is the median of the characters $S_{i - 1}$, $S_i$ and $S_{i + 1}$. We allow the string to be circular, i.e. we consider $S_0$ as $S_N$ and $S_{N + 1}$ as $S_1$. Given a binary string $S$ of length $N$, define a sequence of strings as follows: - $A_0 = S$ - $A_i = f(A_{i