
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
Inside an equilateral triangle with a side length of \(4 \sqrt{3} \text{ cm}\) moves a circle with a diameter of \(1 \text{ cm}\) such that it continuously touches the boundary of the triangle. Calculate the area of the part of the triangle that can never be covered by the moving circle.

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