
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 right triangle \( \triangle ABC \) with \(\angle C = 90^\circ\), \(\angle A = 60^\circ\), and \(AB = 2\), point \(P\) is inside \(\triangle ABC\) such that \(\angle PAB - \angle PBA = 30^\circ\). Establish an appropriate coordinate system and find the equation of the locus of point \(P\).

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