
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: $(2x-1)^n = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$ ($n \in \mathbb{N}^*$, $n$ is a constant).

(1) Find $|a_0| + |a_1| + |a_2| + \ldots + |a_n|$;

(2) We know that the binomial expansion of $(1+x)^n$ is $(1+x)^n = C_n^0 + C_n^1x + C_n^2x^2 + \ldots + C_n^nx^n$. If we differentiate both sides of this equation with respect to $x$, we get: $n(1+x)^{n-1} = C_n^1 + 2C_n^2x + 3C_n^3x^2 + \ldots + nC_n^nx^{n-1}$. Let $x=1$, we can obtain $C_n^1 + 2C_n^2 + 3C_n^3 + \ldots + nC_n^n = n \cdot 2^{n-1}$. Use this method to solve the following problems:

   (i) Find $1a_1 + 2a_2 + 3a_3 + \ldots + na_n$;

   (ii) Find $1^2a_1 + 2^2a_2 + 3^2a_3 + \ldots + n^2a_n$.

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