
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 sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfying $2a_{n+1}+a_{n}=3$ for $n\geqslant 1$, $a_{1}=10$, and $b_{n}=a_{n}-1$. Find the smallest integer $n$ that satisfies the inequality $|{{S_n}-6}|＜\frac{1}{{170}}$.

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