We have designed a divide$-$and$-$conquer algorithm that runs on an input of size $\backslash ($ n $\backslash ).$ This algorithm works by spending $\backslash ($ O$($n$)\backslash )$ time splitting the problem in half, then does a recursive call on each half, then spending $\backslash ($ O$\backslash$left$($n$^\{2\}\backslash$right$)\backslash )$ time combining the solutions to the recursive calls. On small inputs, the algorithm takes a constant amount of time.

$($a$)$ First, write a recurrence relation that corresponds to the time complexity of the above divide$-$and$-$conquer algorithm.

$($b$)$ Then solve the relation to come up with the worst$-$case time taken for the algorithm. Show all your work.