
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 \( a_{n} \) be the coefficient of the linear term in \( x \) of the expansion of \( (\sqrt{x}+3)^{n} \) (with \( n \geqslant 2 \) and \( n \in \mathbb{N} \)). Evaluate the expression \( \frac{2009}{2008}\left(\frac{3^{2}}{a_{2}}+\frac{3^{3}}{a_{3}}+\cdots+\frac{3^{2009}}{a_{2009}}\right) \).

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