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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