
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  $\{a_n\}$  be a sequence of integers satisfying  $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1$ . If  $2000|a_{1999}$ , find the smallest  $n\ge 2$  such that  $2000|a_n$ .

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