
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 $f(x) = x^2 - ax - a$, where $a \in \mathbb{R}$.  
$(1)$ Determine the parity of the function $f(x)$.  
$(2)$ If the function $F(x) = x \cdot f(x)$ has an extremum at $x = 1$ and the equation $F(x) = m$ has $3$ distinct real roots, find the range of real numbers $m$.  
$(3)$ Let $g(x) = -e^x$ ($e$ is the base of natural logarithm). If for any $x_1, x_2 \in [0,e]$ with $x_1 > x_2$, the inequality $|f(x_1) - f(x_2)| < |g(x_1) - g(x_2)|$ holds, find the range of real numbers $a$.

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