
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 equilateral triangle  $\vartriangle ABC$  be inscribed in a circle  $\omega_1$  with radius  $4$ . Consider another circle  $\omega_2$  with radius  $2$  internally tangent to  $\omega_1$  at  $A$ . Let  $\omega_2$  intersect sides  $\overline{AB}$  and  $\overline{AC}$  at  $D$  and  $E$ ,  respectively, as shown in the diagram. Compute the area of the shaded region.
![Image](https://cdn.artofproblemsolving.com/attachments/a/5/8255cdb8b041719d735607da8139aa4016375d.png)

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