
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 for each integer \( m \) satisfying \( 0 < m < 2001 \), there exists an integer \( k \) such that \( \frac{m}{2001} < \frac{k}{n} < \frac{m+1}{2002} \) holds, where \( n \) is a positive integer, determine the smallest possible value of \( n \).

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