Wait. Number Theory?

Calculus Level 5

\[\large\sum _{n=1}^{\infty}\sum _{m=1}^{\infty}\dfrac{1}{(m^2+n^2)^2}=\dfrac{\pi^a}{b}G - \dfrac{\pi^d}{f}\]

The equation above holds true for positive integers \(a,b,d\) and \(f\). Find \(a+b+d+f\).

\[\] Notation: \(\displaystyle G = \sum_{n=0}^\infty \dfrac{ (-1)^n}{(2n+1)^2} \approx 0.916 \) denotes the Catalan's constant.

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