
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 $\{a_n\}$ that satisfies $3a_{n+1} + a_n = 4$ (for $n \geq 1$), and $a_1 = 9$, find the smallest integer $n$ that satisfies the inequality $|S_n - n - 6| < 1$.

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