
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  $n$  be a positive integer with at least  $4$  positive divisors. Let  $d(n)$  be the number of positive divisors of  $n$ . Find all values of  $n$  for which there exists a sequence of  $d(n) - 1$  positive integers  $a_1$ ,  $a_2$ ,  $\dots$ ,  $a_{d(n)-1}$  that forms an arithmetic sequence and satisfies the following condition: for any integers  $i$  and  $j$  with  $1 \leq i < j \leq d(n) - 1$ , we have  $\gcd(a_i , n) \neq \gcd(a_j , n)$ .

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