Binary String on Steroids

A circular binary string is called *good* if it contains at least one character '1' and there is an integer $d$ such that for each character '1', the distance to the nearest character '1' clockwise is $d$. You are given a circular binary string $S$ with length $N$. You may choose some characters in this string and flip them (i.e. change '1' to '0' or vice versa). Convert the given string into a g