
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 $n$ positive integers \( a_{1}, a_{2}, \cdots, a_{n} \) such that \( 1 < a_{1} < a_{2} < \cdots < a_{n} < 2009 \) and the arithmetic mean of any \( n-1 \) of these numbers is an integer, find the maximum value of \( n \).

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