
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
On a circle of length 2013, there are 2013 points marked, dividing it into equal arcs. At each marked point, there is a token. The distance between two points is defined as the length of the shorter arc between these points. For what largest $n$ can the tokens be rearranged such that each marked point still has a token, and the distance between any two tokens, originally separated by no more than $n$, increases?

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