Each term gives equal value?

Algebra Level 3

Given \(z^2+z+1=0,\) find the value of

\[\left(z+\dfrac{1}{z} \right)^2+\left(z^2+\dfrac{1}{z^2}\right)^2+\left(z^3+\dfrac{1}{z^3} \right)^2+\cdots+\left(z^{21}+\dfrac{1}{z^{21}}\right)^2.\]

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