
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
Let $n$ positive integers $a_1, a_2, \ldots, a_n$ satisfy the following conditions: $1 = a_1 < a_2 < \ldots < a_n = 2009$; and the arithmetic mean of any $n-1$ distinct numbers among $a_1, a_2, \ldots, a_n$ is a positive integer. Find the maximum value of $n$.

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