In this problem, you will design an algorithm to compute $2024^$n given n in N as input. In each case, prove$$the correctness of your algorithm, and an asymptotic upper bound on the number of multiplications used.

$1.$ Using n $1$ many multiplications.

$2.$ Using O$($log$2$ n$)$ many multiplications, assuming n is a power of $2,$ i$.$e$.,$ n $=2^$k

$3.$ Using O$($log$2$ n$)$ many multiplications for any n $($not necessarily a power of $2).$