
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 as shown, with centers at points $A$ and $B$. Each sector subtends an angle of $45^\circ$. Determine the area of the overlapping region.

[asy]
draw((0,0)--(7.07,-7.07)--(14.14,0)--(7.07,7.07)--cycle,black+linewidth(1));
filldraw((7.07,7.07)..(10,0)..(7.07,-7.07)--cycle,gray,black+linewidth(1));
filldraw((7.07,7.07)..(4.14,0)..(7.07,-7.07)--cycle,gray,black+linewidth(1));
label("$A$",(0,0),W);
label("$C$",(7.07,7.07),N);
label("$B$",(14.14,0),E);
label("$D$",(7.07,-7.07),S);
label("$45^\circ$",(0,0),2E);
label("$45^\circ$",(14.14,0),2W);
[/asy]

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