The Rich Get Richer

A bag initially contains one red ball and one blue ball. You select one of these balls at random and then place two balls of that same color back in the bag. For instance, if you drew a red ball, then you would put two red balls back into the bag so that the bag would contain two red balls and one blue bag. You repeat this until the bag contains 2016 balls. For the color that has the most balls present in the bag, what is the expected number of balls of this color? That is, after the bag contains 2016 balls, you open it and choose the color that has more balls (or choose either color in case of a tie); what is the expected number of balls of this color?

If the answer can be expressed as \(\dfrac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, enter \(p+q\) as your answer.

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