
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}= \sum_{i=1}^{n}a_{i} \\);

\\((2)\\) Compare the size of \\(S_{n}\\) with \\((n-2)2^{n}+2n^{2}\\), and explain the reason using mathematical induction.

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