
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 power function $f(x)={({m+1})^2}{x^{{m^2}-m-4}}$ is monotonically increasing on $\left(0,+\infty \right)$, and the function $g\left(x\right)=2x+n$.  
$(1)$ Find the value of $m$;  
$(2)$ When $x\in \left[-1,3\right)$, let $f\left(x\right)$ and $g\left(x\right)$ denote the ranges as sets $A$ and $B$ respectively. Let $p:x\in A$, $q:x\in B$. If $p$ is a necessary condition for $q$ to hold, find the range of real number $n$;  
$(3)$ Let $F\left(x\right)=f\left(x\right)-kx+\left(1-k\right)\left(1+k\right)$, and the minimum value of $F\left(x\right)$ on $\left[0,2\right]$ is $-2$. Find the value of real number $k$.

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