
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
Circles centered at $P$ and $Q$ each have radius 3, as shown. Point $R$ is the midpoint of $\overline{PQ}$, and $PR=3\sqrt{3}$. Segments $RS$ and $RT$ are tangent to the circles centered at $P$ and $Q$, respectively, and $\overline{UV}$ is a common tangent. Calculate the area of the shaded region $RSUTV$.

[asy]
unitsize(1cm);
pair P,Q,S,T,U,V,R;
P=(-3.9,0); Q=(3.9,0); S=(-1.95,1.95);
T=(1.95,1.95); U=(-3.9,3); V=(3.9,3);
R=(0,0);
draw(P--Q);
draw(U--V);
draw(R--S);
draw(R--T);
fill(R--T--V--U--S--cycle,gray(0.6));
draw(P--(-3.9,-3));
draw(Q--(3.9,-3));
label("3",(-3.9,-1.5),W);
label("3",(3.9,-1.5),E);
dot(P);
dot(Q);
dot(S);
dot(T);
dot(U);
dot(V);
dot(R);
fill((-3,2.8)..S--U..cycle,white);
fill((3,2.8)..T--V..cycle,white);
label("$P$",P,W);
label("$Q$",Q,E);
label("$S$",S,SW);
label("$T$",T,SE);
label("$U$",U,N);
label("$V$",V,N);
label("$R$",R,S);
draw(Circle(P,3));
draw(Circle(Q,3));
[/asy]

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