
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 $(x+2)^{n}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\ldots+a_{n}(x-1)^{n}(n\in\mathbb{N}^*)$.

$(1)$ Find $a_{0}$ and $S_{n}= \sum\limits_{i=1}^{n}a_{i}$;

$(2)$ Compare the size of $S_{n}$ and $(n-2)3^{n}+2n^{2}$, and explain the reason.

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