
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 monotonically increasing geometric sequence $\{a_n\}$ 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 for the sequence $\{a_n\}$.

(2) If $b_n = a_n \log_{\frac{1}{2}} a_n$ and $S_n = b_1 + b_2 + b_3 + \cdots + b_n$, for any positive integer $n$, $S_n + (n + m) a_{n+1} < 0$ always holds, try to find the range of values of $m$.

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