
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 that the function $f(x)$ is a quadratic function, and the solution set of the inequality $f(x) \geqslant 0$ is $\left[-2,3\right]$, and the minimum value of $f(x)$ on the interval $\left[-1,1\right]$ is $4$.
$(1)$ Find the analytical expression of $f(x)$;
$(2)$ Find the analytical expression of the maximum value $H(t)$ of $f(x)$ on the interval $\left[-2,t\right]$;
$(3)$ Let $g(x) = x + 5 - f(x)$. If for any $x \in (-\infty, -\frac{3}{4}]$, $g(\frac{x}{m}) - g(x-1) \leqslant 4[m^{2}g(x) + g(m)]$ always holds, find the range of real numbers for $m$.

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