
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
Select one vertex of a regular 400-sided polygon. Among the diagonals starting from this vertex, find those for which the absolute value of the difference between the diagonal length and twice its distance from the center differs the least from the radius of the circumscribed circle.

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