
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 $\{a_n\}$ satisfying $2a_1 + a_3 = 3a_2$, and $a_3+2$ is an arithmetic mean between $a_2$ and $a_4$;
(1) Find the general formula for the sequence $\{a_n\}$;
(2) Let $b_n = a_n - \log_2 a_n$, $S_n = b_1 + b_2 + \ldots + b_n$, find the smallest value of $n$ such that the inequality $S_n - 2^{n+1} + 47 < 0$ holds.

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