
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
Let $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$, and it is known that $a_{1}=3$, $S_{n}+n\left(n+1\right)=na_{n+1}$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ If for any $n\in \mathbb{N}^{*}$, $m≥\frac{{a}_{1}}{{3}^{1}}+\frac{{a}_{2}}{{3}^{2}}+⋯+\frac{{a}_{n}}{{3}^{n}}$, find the smallest integer value of $m$.

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