Problem $1(3+5=8)$ What does this code do$?$

You encounter the following mysterious piece of code.

Algorithm $1$: Mystery Function

$1$ Function F$($a$,$ n$))$:

$2$ If n $=0$ :

$3$ Return $(1,$ a$)$

$4$ Else

$5$ b $=1$

$6$ For i from $1$ to $2$n

$7$ b $=$ b $$ a

$8($u$,$ v$)$ F$($a$,$ n $1)$

$9$ Return $($u $$ b$/$a$,$ v $$ b $$ a$)$

$($a$)$ What are the results of F$($a$,3),$ F$($a$,4),$ and F$($a$,5).$ You do not need to justify your answers.

$($b$)$ What does the code do in general, when given input integer n $>=0?$ Prove your assertion by induction

on n$.$