$(1+1+1+3$ points$)$ A Hamiltonian path in a directed graph $G$ is a directed path that goes through each node exactly once. Let

is a directed graph with a Hamiltonian path from $s$ to $t,$ where $s$ and $t$ are two specified nodes in $G\}.$

$($a$)$ Is the language HAMPATH decidable? $($Yes $/$ No $/$ Unknown$)$

$($b$)$ Is the language HAMPATH Turing$-$recognizable? $($Yes $/$ No $/$ Unknown$)$

$($c$)$ Give a definition of the class of languages NP$.$

is a language that

$($d$)$ Show that HAMPATH is in NP$.$

Construct a nondeterministic Turing machine $N$ that

$N$