
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 $3a_{n+1} + a_n = 4$, with $a_1 = 9$, and the sum of its first $n$ terms is denoted by $S_n$. Find the smallest natural number $n$ that satisfies the inequality $|\begin{matrix} S_n - n - 6 \end{matrix}| < \frac{1}{125}$.

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