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I roll two $k$-sided die, and if the same faces appear on both die, which happens with probability $p$, I roll again and again until the faces disagree with probability $(1-p)$ (implying a negative binomial distribution). Let $R$ be the number of successful dice rolls before the faces disagree.

If I repeat the above dice roll process $N$ times, and each time I add $C*R^u$ to a growing sum $Y$, for some arbitrary $C$ and $u$, what is the distribution and mean for $Y$?

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