Bipartite Graphs and Matchings · Challenge problem

A network has two separate pieces with no links between them. The first has workers $u_1,u_2,u_3$ each linkable to either resource $a$ or resource $b$. The second is a ring $1\text{-}2\text{-}3\text{-}4\text{-}5\text{-}6\text{-}1$. How many different maximum independent sets (sets of this maximum possible size with no two chosen vertices linked) does the whole network have?