
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 $\{a_{n}\}$ where $a_{1}=4a_{5}$ and $a_{3}=4a_{6}+6$, and a sequence $\{b_{n}\}$ defined as $b_{n}=a_{n}\cdot a_{n+1}\cdot a_{n+2}$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Determine the smallest value of $n$ for which $S_{n} \lt 0$.

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