
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
Find all positive integers \( n \) such that there exist positive integers \( x_{1}, x_{2}, \cdots, x_{2012} \) satisfying
\[ x_{1} < x_{2} < \cdots < x_{2012} \]
and
\[ \frac{1}{x_{1}} + \frac{2}{x_{2}} + \cdots + \frac{2012}{x_{2012}} = n. \]

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