
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
An octagonal dart board is designed with a regular hexagon at the center and the remaining area divided equally among triangular regions formed between the hexagon and the octagon. If a dart thrown at the board lands anywhere on the board with equal likelihood, what is the probability that the dart lands within the center hexagon?
A) \(\frac{3\sqrt{3}}{8(1+\sqrt{2})}\)  
B) \(\frac{1}{4}\)  
C) \(\frac{1}{8}\)  
D) \(\frac{\sqrt{3}}{8}\)

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