
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 that the sequence $\{a_{n}\}$ is an arithmetic sequence with first term $a_{1}=2$, and $a_{3}$ is the geometric mean of $a_{2}$ and $a_{4}+1$.

(1) Find the general term formula for the sequence $\{a_{n}\}$;

(2) Let $b_{n}=\frac{2}{(n+3)(a_{n}+2)}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$, denoted as $S_{n}$.

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