
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 $(x+1)^n = a_0 + a_1(x-1) + a_2(x-1)^2 + a_3(x-1)^3 + \ldots + a_n(x-1)^n$, (where $n \in \mathbb{N}^*$)  
(1) Find $a_0$ and $S_n = a_1 + 2a_2 + 3a_3 + \ldots + na_n$;  
(2) Compare the magnitude of $S_n$ and $n^3$, and explain the reason.

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