
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 the function $f(x)=ax^{2}+x-15$.  
$(1)$ If the solution set of the inequality $f(x)\leqslant 0$ with respect to $x$ is $\left[2m-8,m\right]$, find the values of $a$ and $m$;  
$(2)$ Under the condition of $(1)$, in the inequality system with respect to $x$: $\left\{\begin{array}{l}{f(x)＞0}\\{2{x}^{2}+(2k+7)x+7k＜0}\end{array}\right.$, the solution set contains exactly two integer solutions. Find the range of values for $k$.

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