
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 $\{a_{n}\}$ that satisfies $\sqrt{a_{1}}+\sqrt{a_{2}}+⋯+\sqrt{a_{n}}=\frac{n^{2}+n+2}{2}$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $b_{n}=\frac{1}{a_{n}+\sqrt{a_{n}}}$, find the sum $S_{n}$ of the first $n$ terms of the sequence $\{b_{n}\}$.

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