
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
Consider that the three vertices of triangle \( \triangle ABC \) in the complex plane correspond to the complex numbers \( z_1, z_2, z_3 \). Given that \( |z_1| = 3 \), \( z_2 = \bar{z}_1 \), and \( z_3 = \frac{1}{z_1} \), find the maximum area of \( \triangle ABC \).

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