$3.$ Consider an undirected graph G that has $\backslash ($ n $\backslash )$ distinct vertices where each vertex has degree $2.$ Assume $\backslash ($ n $\backslash$geq $3\backslash ).$

$($a$.)$ What is the maximum number of distinct circuits that G can contain?

$($b$.)$ If all the vertices of G are contained in a single circuit, what is the maximum number of vertices that can be contained in an independent set?