
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 for some number $x_{0}$ within its domain, the function $f(x)$ satisfies $f(x_{0})=x_{0}$, then $x_{0}$ is called a fixed point of $f(x)$. Given the function $f(x)=ax^{2}+(b+1)x+b-1(a\neq 0)$.

$(1)$ When $a=1$, $b=-2$, find the fixed points of the function $f(x)$;  
$(2)$ If for any real number $b$, the function $f(x)$ always has two fixed points, find the range of values for $a$;  

$(3)$ Under the condition of $(2)$, if the abscissas of two points $A$, $B$ on the graph of $y=f(x)$ are the fixed points of the function $f(x)$, and the midpoint $C$ of $A$, $B$ lies on the graph of $g(x)=-x+\frac{a}{5a^{2}-4a+1}$, find the minimum value of $b$.

Reference formula: The midpoint coordinates of $A(x_{1},y_{1})$, $B(x_{2},y_{2})$ are $\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)$

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