C - Balanced Subsequence

We define a set of **balanced** strings in the following way: - $()$ and the empty string are balanced. - If $A$ is balanced, so is $(A)$. - If $A$ and $B$ are balanced, so is $A + B$. - $A + B$ denotes the concatenation of $A$ and $B$. Given a string $S$, let $f(S) = $ the length of the longest balanced string, which is a subsequence of $S$. Note that a subsequence need not be continuous.