
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
Given functions $f(x)=a{x^2}-({2\sqrt{4+2b-{b^2}}})x$ and $g(x)=-\sqrt{1-{{({x-a})}^2}}$ where $a,b∈{R}$.  
$(1)$ When $b=0$, if $f\left(x\right)$ has a maximum value of $4$, find the value of $a$.  
$(2)$ Find all integer pairs $\left(a,b\right)$ that satisfy the following conditions: there exists $x_{0}$ such that $f(x_{0})$ is the maximum value of $f\left(x\right)$ and $g(x_{0})$ is the minimum value of $g\left(x\right)$.  
$(3)$ For the integer pairs $\left(a,b\right)$ satisfying the conditions in $(2)$, given the domain $D=\{x\left|\right.x\in R$ and $x\neq 2k,k\in Z\}$, the function $h\left(x\right)$ satisfies: $h\left(x+2\right)=-h\left(x\right)$, and when $x\in \left(-2,0\right)$, $h\left(x\right)=f\left(x\right)$. If the number of zeros of the function $y=h\left(x\right)-mx$ is $4$, find the range of real numbers for $m$.

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