$(10$ points$)$ Consider the following problem:

Given a simple directed graph $G$ a starting vertex $s,$ an ending vertex $t$ and one intermediate vertex $v.$ Consider the following algorithm that claims to determine if there exists a path in $G$ from $s$ to $t$ that goes through $v.$

$($In a path, you cannot repeat vertices and you cannot repeat edges.$)$

For each algorithm, Provide a runtime analysis $($Based on $|V|$ and $|E|)$

identify if it correctly solves the problem.

If it is correct, provide a correctness proof. If it is not correct, provide a counterexample. $($a$)$ ; a directed graph $G$ and vertices $s,v,t.$