
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 marked points dividing it into equal arcs. Each marked point has a chip. We call the distance between two points the length of the smaller arc between them. For what largest $n$ can the chips be rearranged so that there is again a chip at each marked point, and the distance between any two chips, initially no more than $n$ apart, increases?

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