
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
If the functions $f(x)$ and $g(x)$ satisfy: for any $x_{1} \in D$, there exists a unique $x_{2} \in D$ such that $f(x_{1})g(x_{2})=m$, then $f(x)$ is called the "$m$-order self-adjoint function" of $g(x)$ on interval $D$; for any $x_{1} \in D$, there exists a unique $x_{2} \in D$ such that $f(x_{1})f(x_{2})=m$, then $f(x)$ is called the "$m$-order self-adjoint function" on interval $D$.  
$(1)$ Determine whether $f(x)=x^{2}+1$ is a "$2$-order self-adjoint function" on the interval $[0,3]$ and explain the reason;  
$(2)$ If the function $f(x)=3x-1$ is a "$1$-order self-adjoint function" on the interval $[\frac{1}{2},b]$, find the value of $b$;  
$(3)$ If $f(x)=\frac{4}{{x+2}}$ is a "$2$-order self-adjoint function" of $g(x)=x^{2}-2ax+a^{2}-1$ on the interval $[0,2]$, find the range of real number $a$.

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