
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 length of the common external tangent of two externally tangent circles is $20 \mathrm{~cm}$. What are the radii of these circles if the radius of the circle that is tangent to both of these circles and their common tangent, placed between them, is as large as possible?

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