
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
Suppose that \(\frac{2009}{2014} + \frac{2019}{n} = \frac{a}{b}\), where \(a\), \(b\), and \(n\) are positive integers with \(\frac{a}{b}\) in lowest terms. What is the smallest positive integer \(n\) for which \(a\) is a multiple of 1004?

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