
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
For a function $y$, if there exists at least one point $A(a, a^{2})$ on the graph of the function, we may call this function a climbing function, and call this point $A$ the climbing star of the function. Please try to complete the following questions based on the above rules:<br/>$(1)$ Try to determine if the function $y=\frac{{-8}}{x}$ is a climbing function. If it is, find the climbing star; if not, explain the reason;<br/>$(2)$ Given a linear function $y=mx+m$ is a climbing function and has a unique climbing star, find the analytical expression of the function and the climbing star;<br/>$(3)$ Given a quadratic function $y=mx^{2}+nx+1$ $(m\geqslant 2)$ is a climbing function, with two climbing stars denoted as ${A_1}({{a_1},a_1^2})$ and ${A_2}({{a_2},a_2^2})$, $|a_{1}-a_{2}|=p$, the minimum value of the function is $q$, and $p^{2}=3q$, find the range of values for $q$.

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