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Algebra Level 5

\[\large x_1x_3+\sum_{k=2}^{2015}x_kx_{k+1}\]

Let \(M\) denote the maximum value of the expression above, where \(x_1, x_2,\ldots, x_{2016} \) are real numbers satisfying the condition \( \displaystyle \sum_{k=1}^{2016} x_k^2 = 1\).

Given that \(M = \cos\left( \dfrac \pi n\right) \) for some positive integer \(n\), find \(n\).


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