
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
(1) Let $(3x-1)^4 = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4$.
   ① Find $a_0 + a_1 + a_2 + a_3 + a_4$;
   ② Find $a_0 + a_2 + a_4$;
   ③ Find $a_1 + a_2 + a_3 + a_4$;
(2) Find the remainder of $S = C_{27}^{1} + C_{27}^{2} + \ldots + C_{27}^{27}$ when divided by 9.

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