Crazy Antique

Algebra Level 5

\[\large{\begin{cases} x+\dfrac{1}{x}&=n \\ x^7+\dfrac{1}{x^7}&=n\left(x^6+\dfrac{1}{x^6}\right) \end{cases}}\]

If \(n_1,n_2,n_3,\dots,n_k\) are the solutions for \(n\) satisfying the above given conditions, evaluate \(\left(\displaystyle\sum_{i=1}^{k} n_i^2\right) + k\).

Clarification: \(x\) can be a complex number.

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