
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
In  $\triangle ABC$ ,  $AB = AC$ . Its circumcircle,  $\Gamma$ , has a radius of 2. Circle  $\Omega$  has a radius of 1 and is tangent to  $\Gamma$ ,  $\overline{AB}$ , and  $\overline{AC}$ . The area of  $\triangle ABC$  can be expressed as  $\frac{a\sqrt{b}}{c}$  for positive integers  $a, b, c$ , where  $b$  is squarefree and  $\gcd (a, c) = 1$ . Compute  $a + b + c$ .

*Proposed by Aaron Lin*

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