Can this be really done?

Calculus Level 5

$\large \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n \cdot 3^m + m \cdot 3^n) }$

If the series above equals to $$\frac { \alpha }{ \beta }$$ for coprime positive integers $$\alpha, \beta$$, find $$\beta - \alpha$$.

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