Expected Number of Draws to Get a Product

A game involves picking a random element of the set $S = \{1,2,3,4,5\}$ and multiplying it to the previous numbers chosen. The game ends only when the running product is divisible by $6$. What is the expected number of picks before the game ends? The answer is in the form $\frac{\alpha}{\beta}$ for coprime integers $\alpha$ and $\beta$. Input the answer as the sum $\alpha +\beta$.