
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 a sequence $\left\{a_{n}\right\}$ with the general term $a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}$ for $n \in \mathbf{N}^{*}$, and let $S_{n}$ be the sum of the first $n$ terms of the sequence, find the number of rational terms in the sequence $S_{1}, S_{2}, \cdots, S_{2009}$.

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