
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
On the segment \( AB = 2R \) taken as the diameter, a circle is constructed. A second circle, whose radius is half the radius of the first circle, is tangent to the first circle from the inside at point \( A \). A third circle is tangent to the first circle from the inside, to the second circle from the outside, and also tangent to the segment \( AB \). Find the radius of the third circle.

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