
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
An equilateral triangle is inscribed in a circle of radius \( R \). The altitudes of the triangle are extended until they intersect with the circle. These points of intersection are connected, forming a new triangle. Calculate the portion of the area of the circle that lies outside these triangles.

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