
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
For the quadratic function $y=mx^{2}+nx+t\left(m\neq 0\right)$, if there exists $x_{0}\in R$ such that $m{x}_{0}^{2}+n{x}_{0}+t={x}_{0}$ holds, then $x_{0}$ is called a fixed point of the quadratic function $y=mx^{2}+nx+t\left(m\neq 0\right)$.  
$(1)$ Find the fixed points of the quadratic function $y=x^{2}-x-3$;  
$(2)$ If the quadratic function $y=2x^{2}-(2+a)x+a-1$ has two distinct positive fixed points $x_{1}$ and $x_{2}$, and $x_{1}$, $x_{2} \gt 0$, find the minimum value of $\frac{{x}_{1}}{{x}_{2}}+\frac{{x}_{2}}{{x}_{1}}$.

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