
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 first term of the sequence $\{a_{n}\}$ is $a_{1}=3$, and the sum of its first $n$ terms is $S_{n}$, with $S_{n+1}=3S_{n}+2n+3$ for $(n\in N^{*})$.<br/>$(1)$ Find the general term formula $a_{n}$ of the sequence $\{a_{n}\}$;<br/>$(2)$ Let $b_{n}=n(a_{n}+1)$, find the sum $T_{n}$ of the first $n$ terms of the sequence $\{b_{n}\}$.

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