
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 $0$, and the sum of its first $n$ terms is $S_{n}$, where $S_{5}=25$, and $a_{1}$, $a_{2}$, $a_{5}$ form a geometric sequence.<br/>$(1)$ Find the general formula for $\{a_{n}\}$;<br/>$(2)$ Let ${b}_{n}=\frac{4n}{{a}_{n}^{2}•{a}_{n+1}^{2}}$, and denote the sum of the first $n$ terms of the sequence $\{b_{n}\}$ as $T_{n}$. If ${T}_{n}≤{m}^{2}+\frac{1}{2}m$ holds for any $n\in N^{*}$, find the range of values for $m$.

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