# Mod of the cubics!

Algebra Level 4

It is known for a complex number $$z=a+ib$$, ($$a,b\in R$$) that

$|z|=|{ z }^{ 2 }|+|{ z }^{ 3 }|$

It is also know that $0<a<\sqrt { \frac { 1 }{ 2 } \left( 3-\sqrt { 5 } \right) }$

Then $$|b|$$ can be expressed as $\sqrt { \frac { -A{ a }^{ 2 }-\sqrt { B } +C }{ D } }$

Here $$A,B,C,D$$ are positive integers, $$B$$ is square free, $$C$$ and $$D$$ are co-prime.

Find $$A+B+C+D$$

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