$.$ Suppose we have the following algorithm:

$1$: Algorithm Cartesian$($A$,$ B$,$ n$)$

$2$: Input: A and B$,$ two n$-$element lists

$3$: Output: The Cartesian product of the two lists: $[[$A$[0],$ B$[0]],[$A$[0],$ B$[1]],...]$

$4$: Let C be an empty list

$5$: for i from $0$ to n $1$ do

$6$: for j from $0$ to n $1$ do

$7$: Add $[$A$[$i$],$ B$[$j$]]$ to the end of C

$8$: return C

$9$: End.

$($a$)$ What is the time complexity using the RAM model $($i$.$e$.,$ directly counting operations$)?$

Make sure you explain your answer in terms of the operations you consider primitive.

$($b$)$ Is this algorithm$$s running time O$($n

$3$

$)?$ Why or why not?

$($c$)$ Is this algorithm$$s running time $\backslash$Theta $($n

$2$

$)?$ Why or why not?

$($d$)$ Is this algorithm$$s running time $($n$)?$ Why or why not?