The hitting set problem takes in a set X$,$ a family F of subsets of X$,$ and an integer k$,$ and returns whether there is some subset of X of size k that overlaps every element of F$.$ For example, if X $=\{1,2,3,4\},$ F $=\{[1,2],[2,3],[3,4],[4,1]\},$ and k $=2,$ the set $\{2,4\}$ is a solution, since $2$ overlaps the first two sets in F $($hits$$ them$)$ and $4$ overlaps the other two. The $3-$CNF Satisfiability problem $(3$SAT$)$ takes in a Boolean expression in $3-$CNF and returns whether that expression is satisfiable. A Boolean expression in $3-$CNF has a set of clauses joined by $$ands$,$ where each clause is $3$ literals joined by $$ors$.$ Answer the following questions about the reduction algorithm below.