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$\left( x + \frac{1}{x} \right)^2 + \left( x^2 + \frac{1}{x^2} \right)^2 + \cdots + \left( x^{27} + \frac{1}{x^{27}} \right)^2$

If $x$ is a complex number satisfying $x^2 + x + 1 = 0$, determine the value of the expression above.

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