
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
Given an acute triangle \( \triangle ABC \). Let \( A' \) be the point symmetric to \( A \) with respect to \( BC \), and let \( O_A \) be the center of the circle passing through \( A \) and the midpoints of segments \( A'B \) and \( A'C \). Define points \( O_B \) and \( O_C \) similarly. Find the ratio of the circumradii of triangles \( \triangle ABC \) and \( \triangle O_A O_B O_C \).

### Answer
\( 6:1 \).

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