
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 the diagram, $PQ$, $RS$, and $TU$ are diameters of a circle with radius 6, where $PQ$ and $RS$ are perpendicular to each other, and $TU$ is at a $45^\circ$ angle to $PQ$. What is the area of the shaded region?
- [asy]
- size(150);
- import graph;
- fill((-1.5,-1.5)--(-1.5,1.5)--(1.5,1.5)--(1.5,-1.5)--cycle,mediumgray);
- fill(Arc((0,0),sqrt(3),45,135)--cycle,mediumgray);fill(Arc((0,0),sqrt(3),225,315)--cycle,mediumgray);
- draw(Circle((0,0),sqrt(3)));
- draw((-1.5,-1.5)--(1.5,1.5)--(1.5,-1.5)--(-1.5,1.5)--cycle);
- label("$P$",(-1.5,1.5),NW); label("$R$",(1.5,1.5),NE); label("$S$",(-1.5,-1.5),SW); label("$Q$",(1.5,-1.5),SE);
- draw((-2.121,0)--(2.121,0)--(0,2.121)--(0,-2.121)--cycle);
- label("$T$",(-2.121,0),W); label("$U$",(2.121,0),E);
- [/asy]

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