
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 triangle is inscribed in a circle. A second circle, concentric with the first, touches one side of the triangle and divides each of the other two sides into three equal parts. Find the ratio of the radii of these circles.

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