
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 dart board is shaped like a regular decagon. Inside the decagon, exactly in the center, is a circle. All sides of the decagon are of length 2 units, and the radius of the circular region is such that it touches the midpoint of each side of the decagon. If a dart is thrown and it lands randomly on the dartboard, what is the probability that it lands inside the circular region?
A) $\frac{\pi}{5 \tan(\pi/10)}$
B) $\frac{\pi}{10 \tan(\pi/5)}$
C) $\frac{\pi}{10 \tan(\pi/10)}$
D) $\frac{\pi}{20 \tan(\pi/10)}$

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