
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 the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$, with $a_1=1$ and $S_n=na_n-3n(n-1)$ $(n\in \mathbb{N}^*)$.

$(1)$ Find the general formula for the term $a_n$ of the sequence $\{a_n\}$;

$(2)$ Is there a positive integer $n$ such that $\frac{S_1}{1}+\frac{S_2}{2}+\frac{S_3}{3}+\cdots +\frac{S_n}{n}-\frac{3}{2}(n-1)^2=2016$? If it exists, find the value of $n$; if not, explain why.

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