
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 $y=mx^{2}-2m^{2}x+n$ $(m\neq 0)$, passing through points $M(x_{1}, y_{1})$ and $N(x_{2}, y_{2})$.  
$(1)$ If $x_{1}=1$, $x_{2}=3$, and $y_{1}=y_{2}$, then the symmetry axis of this parabola is ______.  
$(2)$ When $x_{1}+x_{2} \gt 4$ and $x_{1} \lt x_{2}$, and $y_{1} \lt y_{2}$, then the range of $m$ is ______.

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