A string w of parentheses $($ and $)$ is balanced if it satisfies one of the

following conditions:

$$ w is the empty string.

$$ w $=($x$)$ for some balanced string x

$$ w $=$ x y for some balanced strings x and y

For example, the string

w $=((())()())(()())()$

is balanced, because w $=$ x y$,$ where

x $=((())()())$ and y $=(()())().$

$($a$)$ Describe and analyze an algorithm to determine whether a given string

of parentheses is balanced