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Algebra Level 5

Consider \(k\) positive real numbers: \(a_1,a_2,a_3,\ldots , a_{k}\) such that \(\displaystyle\prod_{i=1}^{k} a_i=2016\) and \(\displaystyle\sum_{i=1}^{k} a_i^{k}=2016 k \) for some positive integer \(k\).

Then, find the value of \(\displaystyle\sum_{i=1}^{2014} \dfrac{a_i}{a_{i+1}+a_{i+2}}\).

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