Prime Power Roots

Algebra Level 4

\[\begin{cases} \alpha^3 \beta^5 = 1 \\ \alpha^7 \beta^2 = 1 \end{cases}\]

Let \(\alpha = \cos \theta_1 + i \sin \theta_1\) and \(\beta = \cos \theta_2 + i \sin \theta_2\) be the complex numbers satisfying the system above, where \(0 < \theta_1\) and \(\theta_2 < \frac{\pi}{2}\).

If \(\frac{\theta_1}{\theta_2} = \frac{a}{b}\), where \(a, b\) are coprime positive integers, compute \(a+b\).

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