
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
Set  $M$  contains  $n \ge 2$  positive integers. It's known that for any two different  $a, b \in M$ ,  $a^2+1$  is divisible by  $b$ . What is the largest possible value of  $n$ ?

*Proposed by Oleksiy Masalitin*

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