
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
Definition: If the equation about $x$ $a_{1}x^{2}+b_{1}x+c_{1}=0(a_{1}\neq 0$, $a_{1}$, $b_{1}$, $c_{1}$ are constants) and $a_{2}x^{2}+b_{2}x+c_{2}=0(a_{2}\neq 0$, $a_{2}$, $b_{2}$, $c_{2}$ are constants), where the coefficients of the quadratic term, linear term, and constant term in the equations satisfy $a_{1}+a_{2}=0$, $b_{1}=b_{2}$, $c_{1}+c_{2}=0$, then these two equations are called "symmetric equations." For example, the "symmetric equation" of the equation $2x^{2}-3x+1=0$ is $-2x^{2}-3x-1=0$. Based on the above content, solve the following problems:  
$(1)$ Write down directly the "symmetric equation" of the equation $x^{2}-4x+3=0$;  
$(2)$ If the equation about $x$ $3x^{2}+\left(m-1\right)x-n=0$ and $-3x^{2}-x=-1$ are "symmetric equations," find the values of $m$ and $n$, and the solutions of $3x^{2}+\left(m-1\right)x-n=0$.

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