
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
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and it is known that $S_n=n^2+n$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ Let $T_{n}=a_{1}\cdot {3}^{a_{1}}+a_{2}\cdot {3}^{a_{2}}+\ldots \ldots +a_{n}\cdot {3}^{a_{n}}$, find $T_{n}$.

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