
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
Given that the vertices \( A, B, C \) of triangle \( \triangle ABC \) in the complex plane correspond to the complex numbers \( 3+2\mathrm{i}, 3\mathrm{i}, 2-\mathrm{i} \) respectively, and that 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}
user