
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 sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and $2S_{n}=n^{2}+n$, where the general term of the sequence $\{b_{n}\}$ is $b_{n}=\frac{n}{3}•{2^{{a_n}}}({n∈{{N}_+}})$.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ 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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