
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
Let  $ABC$  be an equilateral triangle. Denote by  $D$  the midpoint of  $\overline{BC}$ , and denote the circle with diameter  $\overline{AD}$  by  $\Omega$ . If the region inside  $\Omega$  and outside  $\triangle ABC$  has area  $800\pi-600\sqrt3$ , find the length of  $AB$ .

*Proposed by Eugene Chen*

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