
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
Consider the shape formed from taking equilateral triangle  $ABC$  with side length  $6$  and tracing out the arc  $BC$  with center  $A$ .  Set the shape down on line  $l$  so that segment  $AB$  is perpendicular to  $l$ , and  $B$  touches  $l$ .  Beginning from arc  $BC$  touching  $l$ , we roll  $ABC$  along  $l$  until both points  $A$  and  $C$  are on the line.  The area traced out by the roll can be written in the form  $n\pi$ , where  $n$  is an integer.  Find  $n$ .

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