For any integer \( k \), let \( \alpha_{k}=\cos \left(\frac{k \pi}{...
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For any integer \( k \), let \( \alpha_{k}=\cos \left(\frac{k \pi}{7}\right)+i \sin \left(\frac{k \pi}{7}\right) \), where \( i=\sqrt{-1} \). The value of the expression
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\( \frac{\sum_{k=1}^{12}\left|\alpha_{k+1}-\alpha_{k}\right|}{\sum_{k=1}^{3}\left|\alpha_{4 k-1}-\alpha_{4 k-2}\right|} \) is
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