
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 isosceles triangle $ABC$ (with $AB = AC$), side $\overline{AB}$ is tangent to a circle at point $B$; the circle intersects $\overline{BC}$ at point $D$ and point $C$, $D$ being between $B$ and $C$. If $\angle BAC = 50^\circ$ and $\triangle BDC$ forms a $60^\circ$ triangle, what fraction of the area of $\triangle ABC$ lies outside the circle?
A) $\frac{1}{2} - \frac{\pi}{8}$ 
B) $\sin(50^\circ) - \frac{\sqrt{3}\pi}{12}$ 
C) $\sin(50^\circ) - \frac{\sqrt{3}\pi}{24}$ 
D) $\frac{2\sqrt{3}}{3\pi}$ 
E) $\frac{4 - 4\sqrt{3}\pi}{27}$

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