
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 \(\omega_{1}\) and \(\omega_{2}\) be circles with centers \(O_{1}\) and \(O_{2}\), respectively, and radii \(r_{1}\) and \(r_{2}\), respectively. Suppose that \(O_{2}\) is on \(\omega_{1}\). Let \(A\) be one of the intersections of \(\omega_{1}\) and \(\omega_{2}\), and \(B\) be one of the two intersections of line \(O_{1} O_{2}\) with \(\omega_{2}\). If \(A B = O_{1} A\), find all possible values of \(\frac{r_{1}}{r_{2}}\).

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