
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, $PQ$ and $RS$ are diameters of a circle with radius 6. If $PQ$ and $RS$ are perpendicular, what is the area of the shaded region formed by the areas of $\triangle POS$, $\triangle ROQ$, and the sectors $POR$ and $SOQ$?

[asy]
size(120);
import graph;
fill((-1,-1)--(-1,1)--(1,1)--(1,-1)--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,-1)--(1,1)--(1,-1)--(-1,1)--cycle);
label("$P$",(-1,1),NW); label("$R$",(1,1),NE); label("$S$",(-1,-1),SW); label("$Q$",(1,-1),SE);
[/asy]

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