
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
In the complex plane, the vertices of triangle \( \triangle ABC \) correspond to the complex numbers \( 3+2i \), \( 3i \), and \( 2-i \). A moving point \( P \) corresponds to the complex number \( z \). If the equation \( |\bar{z}|^2 + \alpha z + \bar{\alpha} \bar{z} + \beta = 0 \) represents the circumcircle of \( \triangle ABC \), find the complex numbers \( \alpha \) and \( \beta \).

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