Let $G$ be a complete undirected graph in which all edge lengths are either $2$ or $3($clearly$,G$ satisfies the triangle inequality$).$ Give a $\frac{7}{6}-$approximation algorithm for TSP in this special class of graphs.

Hint: Start by finding a minimum $2-$matching in $G.$ A $2-$matching is a subset $S$ of edges so that every vertex has exactly $2$ edges of $S$ incident at it$.$ You can assume that finding such a minimum $2-$matching can be done in polynomial time.