
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
The circle \(\gamma_{1}\) centered at \(O_{1}\) intersects the circle \(\gamma_{2}\) centered at \(O_{2}\) at two points \(P\) and \(Q\). The tangent to \(\gamma_{2}\) at \(P\) intersects \(\gamma_{1}\) at the point \(A\) and the tangent to \(\gamma_{1}\) at \(P\) intersects \(\gamma_{2}\) at the point \(B\), where \(A\) and \(B\) are distinct from \(P\). Suppose \(PQ \cdot O_{1}O_{2} = PO_{1} \cdot PO_{2}\) and \(\angle APB\) is acute. Determine the size of \(\angle APB\) in degrees.

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