
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 the sum of the first $n$ terms of a sequence $\{a_{n}\}$ as $S_{n}$, where $a_{n} \gt 0$, and the sequence $\{a}_{n},{S}_{n},\frac{1}{3}{a}_{n}^{2}$ form an arithmetic progression.<br/>$(1)$ Find the general term formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If $\{b_{n}\}$ is a geometric sequence with $b_{n} \gt 0$, $b_{2}b_{4}=b_{6}=64$, find the sum of the first $n$ terms of the sequence $\{a_{n}\cdot b_{n}\}$ denoted as $T_{n}$.

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