
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
Two sectors of a circle of radius $10$ overlap offset asymmetrically, with centers $P$ and $R$ as shown, and each sector has a central angle of $30^\circ$. Determine the area of the shaded region.

[asy]
import olympiad;
size(300);
pair P = (0, 0), Q = 10*dir(30), R = (20, 0), S = Q + 10*dir(-30);
draw(circle(P, 10));
draw(circle(R, 10));
filldraw(P--arc(P, Q, S)--cycle, lightgray);
filldraw(R--arc(R, S, Q)--cycle, lightgray);
draw((0,0)--(10*cos(30*pi/180),10*sin(30*pi/180))--(20+10*cos(150*pi/180),10*sin(150*pi/180))--cycle);
label("$P$", P, W);
label("$Q$", Q, NE);
label("$R$", R, E);
label("$S$", S, SE);
label("$30^\circ$", (5, 0), 2N);
[/asy]

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