\[\dfrac{\displaystyle \sum_{k=1}^{12} |\alpha_{k+1}-\alpha_k|}{\displaystyle \sum_{k=1}^3 |\alpha_{4k-1}-\alpha_{4k-2}|}\]

For integer \(k\), let \(\alpha_k=\cos \left( \frac{k \pi}{7} \right)+i \sin \left( \frac{k \pi}{7} \right)\), where \(i=\sqrt{-1}\). Evaluate the value of the expression above.

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