
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 $\{a_n\}$ satisfying $a_n \neq 0$, $a_1 = \frac{1}{3}$, and $\frac{a_{n+1}}{a_n} = 1 - 3a_{n+1}$, derive an expression for the sum of the first $n$ terms of the sequence $\{b_n\}$, where $b_n = \frac{3a_n}{n+2}$. Then determine the smallest positive integer $m$ such that the sum of the first $n$ terms of the sequence $\{b_n\}$ satisfies the inequality $S_n < \frac{1}{4}m^2 - \frac{1}{2}m$.

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