
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
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  $s$ .

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