
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
Let \(P\) and \(Q\) be two distinct points on the circumcircle of triangle \(ABC\) such that \( |PA|^2 = |PB| \cdot |PC| \) and \( |QA|^2 = |QB| \cdot |QC| \) (one of the points is on the arc \(\overset{\frown}{AB}\), and the other is on the arc \(\widehat{AC}\)). Find the difference \(\angle PAB - \angle QAC\), if the difference between the angles \(\angle B\) and \(\angle C\) of triangle \(ABC\) is \(\alpha\).

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