
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 sequence $\{\frac{{a}_{n}}{{3}^{n}}\}$ is a constant sequence with $\frac{1}{3}$ as its first term, and $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$.

$(1)$ Find $S_{n}$.

$(2)$ Let the positive integer $m=b_{0}\times 3^{0}+b_{1}\times 3^{1}+\cdots +b_{k}\times 3^{k}$, where $b_{i}\in \{0,1,2\}$, $i$, $k\in \mathbb{N}$. For example, $3=0\times 3^{0}+1\times 3^{1}$, then $b_{0}=0$, $b_{1}=1$; $4=1\times 3^{0}+1\times 3^{1}$, then $b_{0}=1$, $b_{1}=1$. If $f\left(m\right)=b_{0}+b_{1}+\cdots +b_{k}$, find the sum of the first $n$ terms of the sequence $\{S_{n}\cdot f(S_{n})\}$, denoted as $T_{n}$.

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