
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 geometric sequence \{a_n\} with a common ratio \(q > 1\), 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) Let \(b_n = a_n \log_{\frac{1}{2}} a_n\), and let \(S_n = b_1 + b_2 + \ldots + b_n\). Find the smallest positive integer value of \(n\) that makes \(S_n + n \cdot 2^{n+1} > 62\) hold true.

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