
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 functions $y_1=-x^2+(m+2)x-2m+1$ and $y_2=(n+2)x-2n-3$, where $m$ and $n$ are constants and $n\neq -2$. Let the vertex of function $y_1$ be denoted as $P$.  
$(1)$ When $m=0$, point $P$ lies exactly on the graph of function $y_2$. Find the value of $n$.  
$(2)$ As $m$ varies, do all points $P$ lie on a certain parabola? If so, find the equation of this parabola. If not, explain why.  
$(3)$ When $-1 \lt x \lt 2$, it is always the case that $y_2 \lt y_1$. Find the range of values for $m-n$.

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