How to find the lexicographically smallest string by reversing a substring?

I have a string S which consists of a's and b's. Perform the below operation once. Objective is to obtain the lexicographically smallest string.

Operation: Reverse exactly one substring of S

e.g.

1. if S = abab then Output = aabb (reverse ba of string S)
2. if S = abba then Output = aabb (reverse bba of string S)

My approach

Case 1: If all characters of the input string are same then output will be the string itself.

Case 2: if S is of the form aaaaaaa....bbbbbb.... then answer will be S itself.

otherwise: Find the first occurence of b in S say the position is i. String S will look like

aa...bbb...aaaa...bbbb....aaaa....bbbb....aaaaa...
     |
     i   

In order to obtain the lexicographically smallest string the substring that will be reversed starts from index i. See below for possible ending j.

aa...bbb...aaaa...bbbb....aaaa....bbbb....aaaaa...
     |           |               |               |
     i           j               j               j

Reverse substring S[i:j] for every j and find the smallest string. The complexity of the algorithm will be O(|S|*|S|) where |S| is the length of the string.

Is there a better way to solve this problem? Probably O(|S|) solution.

What I am thinking if we can pick the correct j in linear time then we are done. We will pick that j where number of a's is maximum. If there is one maximum then we solved the problem but what if it's not the case? I have tried a lot. Please help.