
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 $f\left(x\right)=x^{2}-2ax-a$, where $a\in R$.  
$(1)$ Determine the parity of the function $y=f\left(x\right)$;  
$(2)$ Let $F\left(x\right)=x\cdot f\left(x\right)$. If the function $y=F\left(x\right)$ has an extremum at $x=2$, and the equation $F\left(x\right)=m$ has $3$ distinct real roots, find the range of real number $m$;  
$(3)$ Let $g\left(x\right)=-e^{x}$ ($e$ is the base of natural logarithm). If for any $x_{1}$, $x_{2}\in \left[0,e\right]$ and $x_{1} \gt x_{2}$, $|f(x_{1})-f(x_{2})| \lt |g(x_{1})-g(x_{2})|$ holds, find the range of real number $a$.

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