
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 an odd function $f(x)$ defined on $\mathbb{R}$ such that when $x > 0$, $f(x) = x^{2} - 2x - 3$.  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ If there exist $x_{1}$, $x_{2} \in [-4,4]$ such that $|f(x_{1}) - f(x_{2})| \geq m$, find the range of real number $m$.

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