
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 rectangle $PQRS$, angle $R$ is trisected by $\overline{RT}$ and $\overline{RU}$, where $U$ is on $\overline{PQ}$, $T$ is on $\overline{PS}$, $QU=8$, and $PT=3$. Find the area of $PQRS$.

[asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
real length = 2 * (8*sqrt(3) - 3), width = 8*sqrt(3);
draw(origin--(length,0)--(length,width)--(0,width)--cycle);
draw((length,width)--(0,3)^^(length,width)--(length - 8,0));
dot("$P$",origin,SW); dot("$Q$",(length,0),SE); dot("$R$",(length,width),NE); dot("$S$",(0,width),NW); dot("$T$",(0,3),W); dot("$U$",(length - 8,0),S);
[/asy]

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