
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 $f(x) = (2x-3)^n$, the sum of the binomial coefficients of its expansion is 512, and $(2x-3)^n = a + a_1(x-1) + a_2(x-1)^2 + \ldots + a_n(x-1)^n$  
(1) Find the value of $a_2$;  
(2) Find the value of $a_1 + a_2 + a_3 + \ldots + a_n$;  
(3) Find the remainder of $f(20) - 20$ divided by 6.

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