
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
In a quarter circle \( A O B \), let \( A_{1} \) and \( B_{1} \) be the first trisection points of the radii from point \( O \). Calculate the radius of a circle that is tangent to the semicircles with diameters \( A A_{1} \) and \( B B_{1} \), and the arc \( A B \). Use the segment \( O A_{1} \) as the unit of measurement. Additionally, construct the center of the circle.

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