
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
In the plane rectangular coordinate system $xOy$, it is known that the parabola $y=x^{2}-2mx+m^{2}-1$ ($m$ is a constant).  
$(1)$ Find the coordinates of the vertex of this parabola (expressed in terms of $m$ algebraically);  
$(2)$ If the points $A(1-2m, y_{1})$ and $B(m+1, y_{2})$ are both on the parabola, and $y_{1} \gt y_{2}$, find the range of values for $m$.

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