
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 $y=f\left(x\right)$ and $y=g\left(x\right)$ satisfy: for any $x_{1}\in D$, there exists a unique $x_{2}\in D$ such that $f(x_{1})g(x_{2})=m$ holds, then $y=f\left(x\right)$ is called the "$m$-order adjoint function" of $g\left(x\right)$ on the interval $D$; when $f\left(x\right)=g\left(x\right)$, it is called the "$m$-order self-adjoint function" on the interval $D$.  
$(1)$ Determine whether $y=f(x)=\log_{2}({x}^{2}+1)$ is a "$2$-order self-adjoint function" on the interval $[0,\sqrt{7}]$. Explain your reasoning.  
$(2)$ If the function $y=f\left(x\right)=4^{x-1}$ is a "$1$-order self-adjoint function" on the interval $[\frac{1}{2},b]$, find the value of $b$.  
$(3)$ If $y=f(x)=\frac{4}{x+2}$ is the "$2$-order adjoint function" of $y=g\left(x\right)=x^{2}-2ax+a^{2}-1$ on the interval $\left[0,2\right]$, find the range of real numbers for $a$.

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