
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 designed as a regular octagon with a side length of $2 + \sqrt{2}$. A square is inscribed such that its vertices touch the midpoint of each side of the octagon. If a dart thrown at the board is equally likely to land anywhere on the board, determine the probability that the dart lands within the inscribed square.

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