
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 that the sum of the binomial coefficients in the expansion of $f(x) = (2x - 3)^n$ is $512$, and $(2x - 3)^n = a_0 + a_1(x - 1) + a_2(x - 1)^2 + \cdots + a_n(x - 1)^n$.

(1) Find the value of $a_2$.
(2) Find the value of $a_1 + a_2 + a_3 + \cdots + a_n$.
(3) Find the remainder when $f(20) - 20$ is divided by $6$.

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