\[\large \sum_{r=1}^{\infty} \frac{1}{e^{\pi r^{2}}} = \frac{1}{2} \left( \frac{\pi^{\frac{1}{a}}}{\Gamma(\frac{b}{a})}-1\right) \]

Given \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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