
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 $f(x)$ exists a real 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 $f(x)=ax^{2}+(2b-1)x+b-2(a\neq 0)$.  
$(1)$ If for any real number $b$, the function $f(x)$ always has two fixed points, find the range of real number $a$;  
$(2)$ Under the condition of $(1)$, 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 $g(x)=-x+\frac{a}{3{a}^{2}-2a+1}$, find the minimum value of the real number $b$.

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