
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
A regular octagon of side length $2$ is inscribed in a circle. Each side of the octagon subtends an arc. For every two consecutive sides of the octagon, reflect this arc, first over one side and then the next, creating a new arc. What is the area of the region bounded by these $8$ newly created arcs after dual reflection?
A) $3 \pi$
B) $2 \left(\frac{1}{\sin(22.5^\circ)}\right)^2 \pi$
C) $3 \left(\frac{1}{\sin(22.5^\circ)}\right)^2 \pi$
D) $4 \left(\frac{1}{\sin(22.5^\circ)}\right)^2 \pi$

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