
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 an equilateral triangle \( DEF \), sides \( \overline{DE} \) and \( \overline{DF} \) are tangent to a circle at \( E \) and \( F \), respectively. The circle's radius \( r \) is half the length of the triangle's side. What fraction of the area of \( \triangle DEF \) lies outside the circle?
A) \( \frac{1}{2} \)
B) \( \frac{4}{3} - \frac{\pi}{3\sqrt{3}} \)
C) \( \frac{\pi}{4} \)
D) \( \frac{2\sqrt{3} - \pi}{3} \)
E) \( \frac{4\sqrt{3} - \pi}{9} \)

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