<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
670
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?
test