
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
Given a dart board shaped as a regular octagon divided into a central square and four congruent right isosceles triangles, one at each corner, where each triangle contains a smaller square inscribed such that its top vertex touches the hypotenuse of the triangle, calculate the probability that the dart thrown at the board lands within the center square.

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