
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 Cartesian coordinate system $xOy$, the parabola $y=ax^{2}+\left(2m-6\right)x+1$ passes through the point $\left(1,2m-4\right)$.  
$(1)$ Find the value of $a$;  
$(2)$ Find the axis of symmetry of the parabola (expressed in terms of $m$);  
$(3)$ Points $(-m$,$y_{1})$,$(m$,$y_{2})$,$(m+2$,$y_{3})$ lie on the parabola. If $y_{2} \lt y_{3}\leqslant y_{1}$, find the range of values for $m$.

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