
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
The sequence $\left\{a_{n}\right\}$ is an arithmetic sequence and satisfies $3a_{5} = 8a_{12} > 0$. The sequence $\left\{b_{n}\right\}$ satisfies $b_{n} = a_{n} \cdot a_{n+1} \cdot a_{n+2}$ for $n \in \mathbf{N}^{*}$, and the sum of the first $n$ terms of $\left\{b_{n}\right\}$ is denoted as $S_{n}$. For what value of $n$ does $S_{n}$ reach its maximum value? Explain the reason.

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