
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
Four complex numbers \(z\) satisfy the equation
\[
z \overline{z}^3 + \overline{z} z^3 = 450,
\]
where both the real and imaginary parts of \(z\) are integers. These complex numbers are plotted in the complex plane. Determine the area of the quadrilateral they form.

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