
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
If the function $f(x)$ has a value range of $y$ in the interval $[{\frac{1}{b},\frac{1}{a}}]$ when $x\in \left[a,b\right]$, then the interval $\left[a,b\right]$ is called a "reverse domain interval" of $f(x)$. Given an odd function $g(x)$ defined on $\left[-2,2\right]$, where $g(x)=-x^{2}+2x$ when $x\in \left[0,2\right]$.  
$(1)$ Find the analytical expression of $g(x)$.  
$(2)$ Find the "reverse domain interval" of the function $g(x)$ within $\left[1,2\right]$.  
$(3)$ Find all "reverse domain intervals" of the function $g(x)$ within its domain.

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