
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 the sum $\sum_{i=1}^{n} i \cdot C_{n}^{i}$, where $C_{n}^{i}$ represents the binomial coefficient. One possible approach is as follows:

Construct the equation: $C_{n}^{0} + C_{n}^{1}x + C_{n}^{2}x^2 + C_{n}^{3}x^3 + \ldots + C_{n}^{n}x^n = (1+x)^n$

Differentiate both sides with respect to $x$ to obtain: $C_{n}^{1} + 2C_{n}^{2}x + 3C_{n}^{3}x^2 + \ldots + nC_{n}^{n}x^{n-1} = n(1+x)^{n-1}$

Setting $x=1$, we have $C_{n}^{1} + 2C_{n}^{2} + 3C_{n}^{3} + \ldots + nC_{n}^{n} = n \cdot 2^{n-1}$

Analogous to the previous calculation method, compute $\sum_{i=1}^{n} i^2 \cdot C_{n}^{i} = $

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