Blue rolls a fair $n$-sided die numbered with integers from $1$ to $n$, and then flips a coin. The coin is weighted to land heads either $\frac{1}{3}$ or $\frac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\frac{1}{15}$, find $n$.

Let's think step by step and output the final answer within \boxed{}.
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