
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, $AB$ and $CD$ are diameters of a circle with radius 6. If $AB$ and $CD$ intersect at a $60^\circ$ angle at the center $O$, calculate the area of the shaded region formed by $\triangle AOB$, $\triangle COD$, and sectors $AOB$, $COD$.

[asy]
size(120);
import graph;
fill((-2,-1)--(0,2)--(2,-1)--cycle,mediumgray);  // triangles
fill(Arc((0,0),sqrt(2),30,90)--cycle,mediumgray);  // sectors
fill(Arc((0,0),sqrt(2),210,270)--cycle,mediumgray);
draw(Circle((0,0),sqrt(2)));
draw((-2,-1)--(0,2)--(2,-1)--cycle);
label("$A$",(-2,1),NW); label("$B$",(2,1),NE); label("$D$",(2,-1),SE); label("$C$",(-2,-1),SW);
[/asy]

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