
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 a quadratic function $y=mx^{2}-4mx+m-2\left(m\neq 0\right)$, and it intersects the $x$-axis at distinct points $M$ and $N$.  
$(1)$ If the graph of the quadratic function passes through the point $A\left(3,0\right)$,  
① Find the expression of the quadratic function and the coordinates of the vertex;  
② Reflect the part of the parabola's graph between $0\leqslant x\leqslant 5$ along the line $x=5$, and denote the original parabola and the reflected part as graph $F$. If the line $y=kx+n$ passes through the point $C\left(15,1\right)$ and intersects graph $F$ at exactly two points, find the range of values for $n$;  
$(2)$ If $m \lt 0$ and $MN\leqslant 4$, find the range of real numbers for $m$.

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