
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
If the 2009 numbers \(a_{1}, a_{2}, \cdots, a_{2009}\) satisfy: 
\(a_{1} = 3\),
\(a_{n}^{2} - \left(\frac{a_{n-1}}{2009} + \frac{1}{a_{n-1}}\right)a_{n} + \frac{1}{2009} = 0\) (for \(n = 2, 3, 4, \cdots, 2009\)), then the maximum possible value of \(a_{2009}\) is

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