
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
Define: If the function $f(x)$ has a number $x_{0}$ in its domain such that $f(x_{0})=x_{0}$, then $x_{0}$ is called a fixed point of $f(x)$. Given the function $f(x)=ax^{2}+(b+1)x+b-1$ $(a\neq 0)$.<br/>$(1)$ When $a=1$ and $b=3$, find the fixed points of the function $f(x)$;<br/>$(2)$ If for any real number $b$, the function $f(x)$ always has two fixed points, find the range of values for $a$;<br/>$(3)$ Under the conditions of $(2)$, if the abscissas of two points $A$ and $B$ on the graph of $y=f(x)$ are fixed points of the function $f(x)$, and the midpoint $C$ of $A$ and $B$ lies on the graph of the function $g(x)=-x+\frac{{2a}}{{5{a^2}-4a+1}}$, find the minimum value of $b$.

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