
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 $2a\_n = a_{n-1} + a_{n+1}$ for $n \geq 2$, with $a\_1 = 1$ and $a\_2 + a\_4 = 10$, find the minimum value of $\frac{2S\_n + 18}{a\_n + 3}$.

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