Complex roots of a 7th degree polynomial

Algebra Level 5

Consider the equation:

\[x^7+x^6+x^5+x^4+x^3+x^2+x+1=0\]

Let \(x_k\) be the \(k^\text{th}\) distinct root of the equation, and let \(\Re(x_k)\) be the real part of that root.

\[\sum_{k=1}^7{\left[\Re(x_k)\right]^2}=\ ?\]

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