
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 the parabola $y=ax^{2}-2ax\left(a\neq 0\right)$.  
$(1)$ Find the coordinates of the vertex of this parabola (expressed in terms of $a$);  
$(2)$ Two distinct points on the parabola are $\left(-1,m\right)$ and $(x_{0},n)$, if $m=n$, directly determine the value of $x_{0}$;  
$(3)$ Points $A(m-1,y_{1})$ and $B(m+2,y_{2})$ lie on the parabola. Is there a real number $m$ such that $y_{1} \lt y_{2}\leqslant -a$ always holds true? If yes, find the range of values for $m$. If not, explain the reason.

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