
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 $(1+x)^n = a_0 + a_1(x-1) + a_2(x-1)^2 + \dots + a_n(x-1)^n$ ($n\in \mathbb{N}^*$):

(1) Find $a_0$ and $S_n = a_1 + a_2 + a_3 + \dots + a_n$;

(2) Compare the magnitude of $S_n$ and $(n-2)2^n + 2n^2$, and explain the reasons.

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