
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
Let the domains of functions $f(x)$ and $g(x)$ be $D_{1}$ and $D_{2}$, respectively. If for every $x_{0}\in D_{1}$, there exist $n$ distinct real numbers $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{n}\in D_{2}$ such that $g(x_{i})=f(x_{0})$ (where $i=1$, $2$, $3$, $\ldots$, $n$, $n\in \mathbb{N}^{+}$), then $g(x)$ is called an "$n$-fold covering function" of $f(x)$.  
$(1)$ Determine whether $g(x)=2\sin (2x-\frac{\pi}{3})$ $(0\leq x\leq 2\pi)$ is a "$4$-fold covering function" of $f(x)=-(\frac{1}{2})^{|x|}$, and explain the reason.  
$(2)$ Given the function $g(x)=\left\{\begin{array}{l}a{x}^{2}+(2a-3)x+1,-2\leq x\leq 1\\{\log}_{2}x,x>1\end{array}\right.$ is a "$2$-fold covering function" of $f(x)={\log}_{2}\frac{{2}^{x}+2}{{2}^{x}+1}$, find the range of real numbers for $a$.

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