
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 1, 1, and \(\ 2 \sqrt{\frac{13-6 \sqrt{3}}{13}}\ \) are arranged such that the triangle formed by the centers of these circles is equilateral with a side length of \(\sqrt{3}\). Find the radius of the circumcircle of a triangle, each vertex of which is the intersection point of two of these circles, and further from the center of the third circle.

Answer: \(4 \sqrt{3}-6\)

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