
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 $n\in\mathbb{N}^{*}$, $n\geqslant 3$, $k\in\mathbb{N}^{*}$.  
(1) Evaluate:  
① $kC_{n}^{k}-nC_{n-1}^{k-1}$;  
② $k^{2}C_{n}^{k}-n(n-1)C_{n-2}^{k-2}-nC_{n-1}^{k-1}$ ($k\geqslant 2$);  
(2) Simplify: $1^{2}C_{n}^{0}+2^{2}C_{n}^{1}+3^{2}C_{n}^{2}+\ldots+(k+1)^{2}C_{n}^{k}+\ldots+(n+1)^{2}C_{n}^{n}$.

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