\[f(p,q)=\displaystyle\prod_{i=0}^{\infty} \left (1+\frac{1}{q^{p^i}}\right )\]

If \(\displaystyle\sum_{n=2}^{\infty} \frac{\phi_1^n-\phi_2^n}{\phi_1^{2n}+\phi_2^{2n}-\phi_2^{n-1}\phi_1^{n-1}(\phi_2^2+\phi_1^2)}=\frac{a}{\sqrt{b}}\) where \(b\) is square free, and \(f(a,b)=\frac{\alpha}{\beta}\), calculate \(\alpha+\beta\).

**Details and Assumptions**

\(\phi_1\) is Golden Ratio which equals \(\frac{1+\sqrt{5}}{2}\)

\(\phi_1\phi_2+1=0\)

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