
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 a semicircle $\widehat{AB}$ with center $C$ and radius $3$ sit on diameter $\overline{AB}$. Point $D$ is located on semicircle $\widehat{AB}$ such that the angle $\angle ADB = 90^\circ$. Extend $\overline{AD}$ to $E$ and $\overline{BD}$ to $F$ such that $E$ and $F$ are on arcs $\widehat{AD}$ and $\widehat{DB}$, respectively, with centers at $A$ and $B$ and both having a radius of $4$. The segment $\overline{EF}$ with end points on these extended arcs forms an angle of $45^\circ$ at $D$. Calculate the area of the enclosed space formed by these constructions, denoted as "smile" $AEFBDA$.
A) $\frac{3\pi}{8}$, B) $\frac{5\pi}{8}$, C) $\pi$, D) $2\pi$

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