
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
Define: For the function $y=g\left(x\right)$, when $x\in \left[a,b\right]$, the set of values of $y$ is $[\frac{1}{b},\frac{1}{a}]$, then the interval $\left[a,b\right]$ is called a "inverse mapping interval" of the function $g\left(x\right)$. Given an odd function $f\left(x\right)$ defined on $\left[-3,3\right]$, when $x\in \left(0,3\right]$, $f(x)=1-\frac{1}{2}|x-1|$. <br/>$(1)$ Find the analytical expression of $f\left(x\right)$; <br/>$(2)$ Find the "inverse mapping interval" of the function $f\left(x\right)$ within $\left[1,3\right]$; <br/>$(3)$ Find all "inverse mapping intervals" of the function $f\left(x\right)$ within its domain.

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