
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 an arithmetic sequence $\{a\_n\}$ with a common ratio $q > 1$, and it satisfies: $a\_2 + a\_3 + a\_4 = 28$, and $a\_3 + 2$ is the arithmetic mean of $a\_2$ and $a\_4$.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If $b\_n = a\_n \log\_{ \frac {1}{2}}a\_n$, $S\_n = b\_1 + b\_2 + … + b\_n$, find the smallest positive integer $n$ such that $S\_n + n \cdot 2^{n+1} > 62$.

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