
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 $\{a\_n\}$ that satisfies $a\_2 + a\_3 + a\_4 = 28$, and $a\_3 + 2$ is the arithmetic mean of $a\_2$ and $a\_4$.
(I) Find the general term formula of the sequence $\{a\_n\}$;
(II) Let $b\_n = a\_n \cdot \log\_2 a\_n$, and its first $n$ terms sum is $S\_n$. If $(n-1)^2 \leqslant m(S\_n - n - 1)$ holds for $n \geqslant 2$, find the range of real number $m$.

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