
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
To study the instantaneous rate of change at a certain point $x$ of the differentiable function $f(x) = (1+x)^n$ ($n \in \mathbb{N}^*$), there are two plans to choose from: ① Directly differentiate to get $f'(x)$, then substitute the $x$-coordinate into the expression of the derivative $f'(x)$; ② First expand $f(x) = (1+x)^n$ using the binomial theorem, differentiate term by term, then substitute the $x$-coordinate into the expression of the derivative $f'(x)$. By synthesizing ① and ②, some identities can be obtained. Using the above method, the identity $C_n^1 + 2C_n^2 + 3C_n^3 + \ldots + nC_n^n = n \cdot 2^{n-1}$ ($n \in \mathbb{N}^*$) can be derived.

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