
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 a function, if there exists a real number $m$ such that when the independent variable of the function is $m$, the function value is also $m$, we call this function an intelligent function, and the point $(m,m)$ on the function is called an intelligent point.  
$(1)$ Determine whether the function $y=2x-3$ is an intelligent function;  
$(2)$ The quadratic function $y=ax^{2}+bx+c$ intersects the $x$-axis at points $A(x_{1},0)$ and $B(x_{2},0)$, and ${x}_{1}{x}_{2}+{x}_{1}+{x}_{2}=-\frac{2}{a}$. If the function is an intelligent function regardless of the value of $b$, find the range of values for $a$;  
$(3)$ Under the premise of question $(2)$, if $C$ and $D$ are intelligent points on the function $y=ax^{2}+bx+c$, and $C$ and $D$ are symmetric with respect to the line $y=kx+\frac{a}{2{a}^{2}+a+1}$, find the minimum value of $b$.

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