
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
There are four complex numbers $z$ such that
\[ z \overline{z}^3 + \overline{z} z^3 + z \overline{z} = 500,\]
and both the real and imaginary parts of $z$ are integers. These four complex numbers are plotted in the complex plane. Find the area of the quadrilateral formed by these numbers as vertices.

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