
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 two circles \(\odot O_1\) and \(\odot O_2\) that are externally tangent, with radii \(r_1\) and \(r_2\) respectively (\(r_1 < r_2\)), \(AB\) and \(XY\) are two internal common tangents to the circles, where point \(A\) and point \(X\) lie on \(\odot O_1\), and point \(B\) and point \(Y\) lie on \(\odot O_2\). The circle with \(AB\) as its diameter intersects \(\odot O_1\) and \(\odot O_2\) at points \(P\) and \(Q\) respectively. Given that \(\angle A O_1 P + \angle B O_2 Q = 180^\circ\), find the value of \(\frac{PX}{QY}\) in terms of \(r_1\) and \(r_2\).

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