$(5$ points$)$ Consider the following pivot selection schemes and discuss the complexity for random inputs, pre$-$sorted inputs, and reverse$-$order inputs.

a$.$ Scheme $1$: Pick the first element as a pivot

b$.$ Scheme $2$: Pick the last element as a pivot

c$.$ Scheme $3$: Pick the center element as a pivot

d$.$ Scheme $4$: Median$-$of$-$three pivot selection

$(5$ points$)$ If you had to recommend two pivot selection schemes from the list provided in the previous problem, which ones would you choose? Please explain your reasoning for your selection. If you had to pick just one pivot selection approach from the list, which would it be and why?

$(5$ pts$)$ Professor X asserts that his doctoral student has devised a novel sorting algorithm capable of sorting six numbers through a mere $10$ comparisons. It is incumbent upon you to either support or challenge Professor X$\text{'}$s assertion. What conclusion do you reach, and what are the reasons underlying your decision?

Your decision:

The reasons: