
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 common external tangent of two externally tangent circles makes an angle \( \alpha \) with the line joining their centers. Find the ratio of the radii of these circles.

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