
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
One of two circles is fixed, while the radius of the other circle can vary, but its center remains fixed.

How should we choose the radius of the variable circle so that the two circles have a common internal tangent, and the tangent point on the variable circle is as far as possible from the common symmetry axis of the two circles?

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