
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 $n, k \in \mathbb{N}^*$, and $k \leq n$, $C_n^k = C_{n-1}^{k-1}$, it can be deduced that  
$C_n^1 + 2C_n^2 + 3C_n^3 + \ldots + kC_n^k + \ldots + nC_n^n = n(C_{n-1}^0 + C_{n-1}^1 + \ldots + C_{n-1}^{k-1} + \ldots + C_{n-1}^{n-1}) = n \cdot 2^{n-1}$.  
From this, it can be deduced that $C_n^1 + 2^2C_n^2 + 3^2C_n^3 + \ldots + k^2C_n^k + \ldots + n^2C_n^n = \_\_\_\_\_\_$.

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