Solving complex equation with absolute: $|z|^3 = z^3 + 1$

Question

I've come across an equation: $|z|^3 = z^3 + 1, \quad z \in \mathbb{C}$ and got stuck.
Best I could do was conversion: $$z \overline{z}|z|=(z+1)(z^2 - z + 1)$$ but I don't see an further step for removing final the absolute.

Answer 1

Let $c=z^{3}$. The equation becomes $|c|=c+1$. This makes $c$ real. Since $c=c+1$ has no solution we must have $-c=c+1$ or $c=-\frac 1 2$. Now take cube roots of $-\frac 1 2 $.