
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 graph of the quadratic function $y=f(x)$ intersects the line $y=-6$ at only one point, and satisfies $f(0)=f(-4)=-2$, $g(x)=\frac{{f(x)}}{x}$.  
$(1)$ Find the analytical expression of the quadratic function $y=f(x)$;  
$(2)$ If for any $x\in [1,2]$, $t\in [-4,4]$, $g(x)\geqslant -m^{2}+tm$ always holds, find the range of the real number $m$.

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