
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 an arithmetic sequence \(\{a_n\}\) such that \(3a_5 = 8a_{12} > 0\); another sequence \(\{b_n\}\) is defined by \(b_n = a_n a_{n+1} a_{n+2}\) for \(n \in \mathbb{Z}_+\). Let \(S_n\) denote the sum of the first \(n\) terms of the sequence \(\{b_n\}\). Find the value of \(n\) for which \(S_n\) is maximized. Provide a justification for your answer.

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