
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
A semicircle $\widehat{PQ}$ with center $R$ and radius $2$ contains a point $S$ on $\widehat{PQ}$ such that $\overline{RS}\perp\overline{PQ}$. Extend $\overline{PS}$ and $\overline{QS}$ to points $T$ and $U$, respectively, so that semicircles $\widehat{PT}$ and $\widehat{QU}$ have $Q$ and $P$ as their respective centers. Semicircle $\widehat{TU}$ has center $S$. Calculate the area of the region enclosed by $\widehat{PT}$, $\widehat{TU}$, $\widehat{QU}$, and chord $\overline{PQ}$.
A) $9\pi - 1$
B) $9\pi - 2$
C) $9\pi + 2$
D) $8\pi - 2$

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