
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 arithmetic sequence $\{a_n\}$ with a common difference not equal to $0$, satisfying: $a_1=2$, and $a_1$, $a_2$, $a_5$ form a geometric sequence.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $b_n=\frac{8}{a_n \cdot a_{n+1}}$, and the sum of the first $n$ terms of the sequence $\{b_n\}$ be $S_n$. When $x \in [2,4]$, for any positive integer $n$, the inequality $x^2+mx+m \geqslant S_n$ always holds, find the range of values for $m$.

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