
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
For a positive sequence $\{a_{n}\}$, the sum of the first $n$ terms is $S_{n}$, and $2{S_n}=a_n^2+{a_n}(n\in N^{*})$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ Let ${b_n}=\frac{{{a_1}}}{{{3^n}}}+\frac{{{a_2}}}{{{3^{n-1}}}}+\frac{{{a_3}}}{{{3^{n-2}}}}+⋯+\frac{{{a_{n-1}}}}{{{3^2}}}+\frac{{{a_n}}}{3}$, 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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