
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 $f(x) = mx^{2} + 2mx - 1$ $(m > 0)$.  
$(1)$ If the solution set of the inequality $f(x) < 0$ with respect to $x$ is $(-3, 1)$, and when $x > 2$, find the minimum value of $g(x) = \frac{3f(x) + 4}{x-2}$.  
$(2)$ If there exist $x_{1}, x_{2} \in [-3, 0]$, such that $|f(x_{1}) - f(x_{2})| \geq 4$, find the range of real number $m$.

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