
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
Equilateral triangle  $ABC$  has side length  $2$ . A semicircle is drawn with diameter  $BC$  such that it lies outside the triangle, and minor arc  $BC$  is drawn so that it is part of a circle centered at  $A$ . The area of the “lune” that is inside the semicircle but outside sector  $ABC$  can be expressed in the form  $\sqrt{p}-\frac{q\pi}{r}$ , where  $p, q$ , and  $ r$  are positive integers such that  $q$  and  $r$  are relatively prime. Compute  $p + q + r$ .
![Image](https://cdn.artofproblemsolving.com/attachments/7/7/f349a807583a83f93ba413bebf07e013265551.png)

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