
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 $y=mx^{2}-4mx+3m$.  
$(1)$ Find the coordinates of the intersection points of the graph of this quadratic function with the $x$-axis.  
$(2)$ If $m \lt 0$ and the maximum value of $y$ is $2$ when $1\leqslant x\leqslant 4$, find the minimum value of $y$ when $1\leqslant x\leqslant 4$.  
$(3)$ Given points $P(2$,$\frac{m+4}{2}$) and $Q(4$,$\frac{m+4}{2})$ in the Cartesian coordinate system, when the parabola has a common point with the line segment $PQ$, determine the range of values for $m$.

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