Purify It

We say that a binary string (a string containing only characters '0' and '1') is *pure* if it does not contain either of the strings "0101" or "1010" as a subsequence. Recall that string T is a subsequence of string S if we can delete some of the letters of S (possibly none) such that the resulting string will become T. You are given a binary string $S$ with length $N$. We want to make this