
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
Nine distinct positive integers are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of  $n$  and the product of any two adjacent numbers in the circle is not a multiple of  $n$ , where  $n$  is a fixed positive integer. Find the smallest possible value for  $n$ .

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