
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 $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and ${a}_{n}=\left\{\begin{array}{l}5,n=1\\ 2n+2,n≥2\end{array}\right.$.<br/>$(1)$ Find $S_{n}$;<br/>$(2)$ If ${b}_{n}=\frac{1}{{S}_{n}+1}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$, denoted as $T_{n}$.

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