
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 the geometric sequence \( \{a_n\} \), it is known that \( a_4 = 8a_1 \) and \( a_1 \), \( a_2+1 \), \( a_3 \) form an arithmetic sequence.  
(1) Find the general term formula for the sequence \( \{a_n\} \);  
(2) Calculate the sum of the first \( n \) terms, \( S_n \), for the sequence \( \{ |a_n - 4| \} \).

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