
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
In the arithmetic sequence $\{a\_n\}$, $a\_1^2 + a\_3 = 4$ and $a\_5 + a\_6 + a\_7 = 18$.

1. Find the general term formula of the sequence.
2. If $a\_1$, $a\_2$, $a\_4$ form a geometric sequence, find the sum of the first $n$ terms, $S\_n$, of the sequence $\{ \frac{1}{(2n+2)a\_n} \}$.

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