Eulerian and Hamiltonian Graphs · Hard problem

Two pentagons (five-cycles) share exactly one common vertex $c$: one pentagon is $c\text{-}a_1\text{-}a_2\text{-}a_3\text{-}a_4\text{-}c$ and the other is $c\text{-}b_1\text{-}b_2\text{-}b_3\text{-}b_4\text{-}c$, with no other shared vertices or edges ($9$ vertices, $10$ edges total). How many Hamiltonian cycles does this graph have?