
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For all complex numbers $z$, let 
\[f(z) = \left\{
\begin{array}{cl}
z^{3}&\text{ if }z\text{ is not real}, \\
-z^3 &\text{ if }z\text{ is real}.
\end{array}
\right.\] 
Find $f(f(f(f(1+2i))))$.

Present the answer in LaTex format: \boxed{Your answer}
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