
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 sequence \(\{a_{n}\}\) that satisfies \(3a_{n+1} + a_{n} = 4 \ (n \geq 1)\) and \(a_1 = 9\), and whose sum of the first \(n\) terms is \(S_{n}\), find the smallest integer \(n\) that satisfies the inequality \(\left|S_{n} - n - 6\right| < \frac{1}{125}\).

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