
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
Three circles with radii \( \sqrt{13}, \sqrt{13}, \) and \( 2 \sqrt{13 - 6 \sqrt{3}} \) are arranged such that the triangle formed by their centers is equilateral with a side length of \( \sqrt{39} \). Find the radius of the circumscribed circle around the triangle, each vertex of which is the intersection point of two of these circles, farthest from the center of the third circle.

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