
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 a sequence $\{a_n\}$ satisfying $3a_{n+1} + a_n = 4$ ($n \geq 1$) and $a_1 = 9$, let $S_n$ be the sum of the first $n$ terms. Find the smallest integer $n$ such that 
$$|S_{n} - n - 6| < \frac{1}{125}.$$

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