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Let \( S \) be the set of all \( (\alpha, \beta), \pi\alpha, \beta2 \pi \), for which the complex number \( \frac{1-i \sin \alpha}{1+2 i \sin \alpha} \) is purely imaginary and \( \frac{1+i \cos \beta}{1-2 i \cos \beta} \) is purely real. Let \( Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S \)
Then \( \sum_{(\alpha, \beta) \in S}\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right) \) is equal to:
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