Optimal Expected Value in a Marble Game

Triton performs an ancient Neptunian ritual consisting of drawing red, green, and blue marbles from a bag. Initially, Triton has 3 marbles of each color, and the bag contains an additional 3 marbles of each color. Every turn, Triton picks one marble to put into the bag, then draws one marble uniformly at random from the bag (possibly the one he just discarded). The ritual is completed once Triton has 6 marbles of one color and 3 of another. Compute the expected number of turns the ritual will take, given that Triton plays optimally to minimize this value. The answer is in the form $\frac{\alpha}{\beta}$ for coprime integers $\alpha$ and $\beta$. Input the answer as the sum $\alpha +\beta$.