
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 that  $ \{a_n\}$  is a sequence in which all the terms are integers, and  $ a_2$  is odd. For any natural number  $ n$ ,  $ n(a_{n \plus{} 1} \minus{} a_n \plus{} 3) \equal{} a_{n \plus{} 1} \plus{} a_n \plus{} 3$ . Furthermore,  $ a_{2009}$  is divisible by  $ 2010$ . Find the smallest integer  $ n > 1$  such that  $ a_n$  is divisible by  $ 2010$ .

P.S.: I saw EVEN instead of ODD. Got only half of the points.

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