a$)$ Purely by looking at the code, and NOT running it$,$ what will the return value be when nums is $\{1,2,3,4,5,4,2\}$ and n is $7?$ Explain how you figured out the answer.

b$)$ When passed an array in which the duplicated value occurs in the last two elements, the function runs fine, but when the duplicated value occurs in the first two elements the function starts to run extremely slowly as n gets bigger. Explain the difference in outcomes, based on the how the recursive calls are are placed ic$)$ Consider the case in which the duplicated value is in the last two elements. Derive a recurrence for the number of item comparisons $($red code$).$ Then solve your recurrence in the usual way.

d$)$ Same as part c$,$ but the duplicated value occurs in the first two elements.

e$)$ Optimize the code in the obvious way $($factor out unnecessary recursive call using a temp$).$ Now repeat Part d$.$n the code, and how they end up being called.