
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 $\sum_{i=1}^{n} i \binom{n}{i} = C_{n}^{1} + 2C_{n}^{2} + 3C_{n}^{3} + \ldots + nC_{n}^{n}$. One method to solve this problem is to construct the identity $(1+x)^{n} = C_{n}^{0} + C_{n}^{1}x + 2C_{n}^{2}x^{2} + 3C_{n}^{3}x^{3} + \ldots + nC_{n}^{n}x^{n}$. Taking the derivative with respect to $x$ on both sides yields $n(1+x)^{n-1} = C_{n}^{1} + 2C_{n}^{2}x + 3C_{n}^{3}x^{2} + \ldots + nC_{n}^{n}x^{n-1}$. Setting $x=1$ in this equation gives $C_{n}^{1} + 2C_{n}^{2} + 3C_{n}^{3} + \ldots + nC_{n}^{n} = n \cdot 2^{n-1}$. Using a similar method, compute the sum $\sum_{i=1}^{n} i^2 \binom{n}{i} = C_{n}^{1} + 2^2C_{n}^{2} + 3^2C_{n}^{3} + \ldots + n^2C_{n}^{n} = \_\_\_\_\_\_\_\_.$

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