
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 increasing geometric sequence $\{a_n\}$ such that $a_2+a_3+a_4=28$, and $a_3+2$ is the arithmetic mean of $a_2$ and $a_4$,  
(I) Find the general formula for the sequence $\{a_n\}$;  
(II) Let $b_n = a_n \log_{\frac{1}{2}} a_n$ and $S_n = b_1 + b_2 + \ldots + b_n$, 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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