# 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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