
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
Given is the sequence  $(a_n)_{n\geq 0}$  which is defined as follows: $a_0=3$  and  $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$ .

Determine all positive integers  $m$  such that  $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$ .

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