
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]


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Let $\{a_{n}\}$ be a sequence of positive terms with the sum of the first $n$ terms denoted as $S_{n}$, and ${a}_{n}^{2}+2a_{n}=4S_{n}-1$ for $(n\in N^{*})$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If $b_{n}=3^{n-1}$, find the sum of the first $n$ terms of the sequence $\{\frac{{a}_{n}}{{b}_{n}}\}$ denoted as $T_{n}$.

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