
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 $\{a_n\}$ satisfies $2a_1+a_3=3a_2$, 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) If $b_n=a_n+\log_2 \frac{1}{a_n}$, $S_n=b_1+b_2+\ldots+b_n$, find the smallest positive integer $n$ for which $S_n-2^{n+1}+47 < 0$ holds.

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