
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 polygon on a chessboard. Suppose a chessboard consists of squares with a side length of 4. Furthermore, suppose a regular $4n$-gon with a "radius" of $1^*$ is thrown onto this board. Determine the probability that this polygon will intersect the side of any square.

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