An algebra problem by Kshetrapal Dashottar

Algebra Level 4

\[\begin{cases} (x^2 + xy + y^2)(y^2 + yz + z^2)(z^2 + zx + x^2) = xyz \\ \\ (x^4 + x^2y^2 + y^4)(y^4 + y^2z^2 + z^4)(z^4 + z^2x^2 + x^4) = x^3y^3z^3 \end{cases} \]

Non-zero real numbers \(x\), \(y\) and \(z\) satisfy the system of equations above. Find \((xyz)^{-1}\).

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