
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 natural number. The pairwise distinct nonzero integers  $a_1,a_2,\ldots,a_n$  have the property that the number \[(k+a_1)(k+a_2)\cdots(k+a_n)\]is divisible by  $a_1a_2\cdots a_n$  for any integer  $k{}.$  Find the largest possible value of  $a_n.$ *Proposed by F. Petrov and K. Sukhov*

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