Too Complex to Solve!

Algebra Level 3

Let \( z=a+bi\), \(|z|=5\), and \(b>0\). If the distance between \((1+2i)z^3\) and \(z^5\) is as large as possible, then determine \(z^4\). Give your answer as the sum of the real and imaginary parts of \(z^4\).

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