
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
Right  $\vartriangle ABC$  has side lengths  $6, 8$ , and  $10$ . Find the positive integer  $n$  such that the area of the region inside the circumcircle but outside the incircle of  $\vartriangle ABC$  is  $n\pi$ .
![Image](https://cdn.artofproblemsolving.com/attachments/d/1/cb112332069c09a3b370343ca8a2ef21102fe2.png)

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