For a regular hexahedron and a regular octahedron, all their faces are regular triangles with each side of length $a$. Let their inradii be $r_1$ and $r_2$, respectively. If $\frac{r_1}{r_2} = \frac{m}{n}$, where $\gcd(m, n) = 1$, then find the value of $mn$.

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