A string containing just the characters $($ and $)$ is valid if the parentheses are well matched. For example $(())()$ and $((()))$ are valid, while $(()$ and $)($ are not. Also recall that a subsequence consists of a subset of characters of the string in the original order but not necessar$-$ ily consecutive, while a substring must consist of consecutive characters. Given a string s with n characters of $($ and $),$ design an algorithm for the following tasks: