
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

Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\). Segment \(PQ\) is tangent to \(\omega_{1}\) at \(P\) and to \(\omega_{2}\) at \(Q\), and \(A\) is closer to \(PQ\) than \(B\). Point \(X\) is on \(\omega_{1}\) such that \(PX \parallel QB\), and point \(Y\) is on \(\omega_{2}\) such that \(QY \parallel PB\). Given that \(\angle APQ=30^{\circ}\) and \(\angle PQA=15^{\circ}\), find the ratio \(AX / AY\).

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