To minimize the number of operations, we want to perform more operations on characters where \(S_i \neq T_i\).

Let’s take out only the characters where \(S_i \neq T_i\) from \(T\) from left to right, and create a string \(U\) by replacing \(T_i\) with ( if it’s A, and with ) if it’s B.

For example, if \(S\) is BAABA and \(T\) is AABBB, then \(U\) will be ()).

Now, if \(U\) is a valid sequence of parentheses, then the answer will be half of \(|U|\). We just need to perform operations that correspond to the matching pairs in the parentheses sequence.

Otherwise, we need to determine the minimum number of ( to insert at the leftmost position where \(T_i\) is A, and the minimum number of ) to insert at the rightmost position where \(T_i\) is B, in order to make it a valid sequence of parentheses. (There are cases where it is not possible to make it a valid sequence of parentheses depending on the leftmost position where \(T_i\) is A or the rightmost position where \(T_i\) is B, and in those cases, the answer is \(-1\).)

The minimum number of insertions needed can be determined greedily by inserting as many ( as needed at the beginning and then as many ) as needed at the end.