
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
In the sequence $\{a_{n}\}$, given that ${a_1}=\frac{1}{2}$, ${a_{n+1}}=({1+\frac{1}{n}}){a_n}+\frac{{n+1}}{{{3^n}}}$.  
$(1)$ Let ${b_n}=\frac{{{a_n}}}{n}$, find the general formula for the sequence $\{b_{n}\}$;  
$(2)$ Find the sum of the first $n$ terms $S_{n}$ of the sequence $\{a_{n}\}$.

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