If only it were as easy as 1, 2, 3

Algebra Level 2

If zz is a complex number satisfying the equation

z+1z=z2+1z2<0,z+\frac{1}{z}=z^2+\frac{1}{z^2}<0,

what is the value of z3+1z3z^3+\frac{1}{z^3}?

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