
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 the function $f(x)$, if for any given real number in its domain $D$, we have $-x \in D$, and $f(x)f(-x) = 1$, then $f(x)$ is called the "inverse function".  
$(1)$ Determine whether the function $f(x) = \frac{8+x^3}{8-x^3}$ ($x^2 \neq 4$) is an "inverse function", and explain the reason;  
$(2)$ If the graph of the inverse function $g(x)$ with domain $R$ is a continuous curve, and $g(x)$ is monotonically increasing on $(-\infty, 0)$ with $g(x) > 0$ for all $x \in (-\infty, 0)$.  
① According to the definition, study the monotonicity of $g(x)$ on $R$;  
② If $g(-2) = \frac{1}{2}$, and the function $h(x) = [g(x)]^2 + [g(-x)]^2 - g(x) - g(-x)$, find the range of $h(x)$ on $[-2, 2]$.

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