
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
A circle is constructed on the segment \( AB = 2R \) as on a diameter. A second circle of the same radius as the first has its center at point \( A \). A third circle touches the first circle internally, the second circle externally, and also the segment \( AB \). Find the radius of the third circle.

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