The arithmetic mean of $a$ and $b$ is $\frac{a+b}{2}$ (the average). The harmonic mean of $a$ and $b$ is $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ (the reciprocal of the average of the reciprocals). There is a unique positive integer $n$ such that $n$ and $5$ are distinct positive integers whose arithmetic mean and harmonic mean are both integers. Compute $n$.

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