<|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
 Let  $n \ge 2$  be an integer. Alex writes the numbers  $1, 2, ..., n$  in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment  $s$  we denote by  $d_s$  the difference of the numbers written in its extremities and by  $p_s$  the number of all other drawn segments which intersect  $s$  in its interior.
Find the greatest  $n$  for which Alex can write the numbers on the circle such that  $p_s \le |d_s|$ , for each drawn segment  $s$ . 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
11
Let  $n \ge 2$  be an integer. Alex writes the numbers  $1, 2, ..., n$  in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment  $s$  we denote by  $d_s$  the difference of the numbers written in its extremities and by  $p_s$  the number of all other drawn segments which intersect  $s$  in its interior.
Find the greatest  $n$  for which Alex can write the numbers on the circle such that  $p_s \le |d_s|$ , for each drawn segment  $s$ .
test