
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
Semicircle $\widehat{AB}$ has center $C$ and radius $r$. Point $D$ is on $\widehat{AB}$ such that $\overline{CD}\perp\overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to points $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers with radius $2r$. Circular arc $\widehat{EF}$ has center $D$. Calculate the area of the region $AEFBDA$.
A) $\pi r^2 - 4r^2$
B) $2\pi r^2 - 2r^2$
C) $\pi r^2 - 2r^2$
D) $2\pi r^2 - 4r^2$
E) $3\pi r^2 - 3r^2$

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