
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
The sequence of integers \( u_{0}, u_{1}, u_{2}, u_{3}, \cdots \) satisfies \( u_{0}=1 \), and for each positive integer \( n \), \( u_{n+1} u_{n-1}=k u_{n} \), where \( k \) is some fixed positive integer. If \( u_{2000}=2000 \), find all possible values of \( k \).

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