
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 $\left\{a_{n}\right\}$ satisfies $3 a_{n+1}+a_{n}=4$ for $n \geqslant 1$, and $a_{1}=9$, with the sum of the first $n$ terms being $S_{n}$, find the smallest integer $n$ such that $\left|S_{n}-n-6\right|<\frac{1}{125}$.

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