
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
Calculate $C_n^1 + 2C_n^2 + 3C_n^3 + \ldots + nC_n^n$. The following method can be used: Construct the equation $C_n^0 + C_n^1x + C_n^2x^2 + \ldots + C_n^nx^n = (1+x)^n$. Differentiate both sides with respect to $x$, obtaining $C_n^1 + 2C_n^2x + 3C_n^3x^2 + \ldots + nC_n^nx^{n-1} = n(1+x)^{n-1}$. Substituting $x=1$ into the above equation, we get $C_n^1 + 2C_n^2 + 3C_n^3 + \ldots + nC_n^n = n \cdot 2^{n-1}$. By analogy with the above calculation method, calculate $C_n^1 + 2^2C_n^2 + 3^2C_n^3 + \ldots + n^2C_n^n$.

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