
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 circles \( S_1 \) and \( S_2 \) with centers \( O_1 \) and \( O_2 \) respectively touch each other externally; a line touches the circles at distinct points \( A \) and \( B \) respectively. It is known that the intersection point of the diagonals of quadrilateral \( O_1 A B O_2 \) lies on one of the circles. Find the ratio of the radii of the circles.

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