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

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

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