
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
In the arithmetic sequence $\{a_{n}\}$, $a_{2}=-1$, and the sum of the first $n$ terms of $\{a_{n}\}$ is $S_{n}$, satisfying $S_{9}=5S_{5}$.  
$(1)$ Find the general formula for the arithmetic sequence $\{a_{n}\}$;  
$(2)$ If ${b}_{n}=\frac{1}{{a}_{n+1}•{a}_{n+2}}$, and $T_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$, and there exist constants $s$ and $t$ such that the inequality $s\leqslant T_{n}\leqslant t$ holds for any positive integer $n$, find the minimum value of $t-s$;  
$(3)$ If for any $n\in N$, $n\geqslant 3$, the inequality $\frac{{S}_{n}}{{a}_{n}}≤\frac{{n}^{2}}{n+k}$ holds, find the maximum value of the positive number $k$.

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