
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 $mx_{0}^{2}+nx_{0}+t=x_{0}$ 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 $y=2x^{2}-(3+a)x+a-1$ has two distinct positive fixed points $x_{1}$ and $x_{2}$, find the minimum value of $\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1} }$;  
$(3)$ If for any real number $b$, the quadratic function $y=ax^{2}+(b+1)x+(b-1)\left(a\neq 0\right)$ always has a fixed point, find the range of values for $a$.

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