
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 graphs of the functions $y=f(x)$ and $y=g(x)$ each have points $M$ and $N$ that are symmetric about the $x$-axis, then the functions $y=f(x)$ and $y=g(x)$ are said to have a $C$ relationship.  
$(1)$ Determine whether the functions $f(x)=\log_{2}(8x^{2})$ and $g(x)=\log_{\frac{1}{2}}x$ have a $C$ relationship;  
$(2)$ If the functions $f(x)=a\sqrt{x-1}$ and $g(x)=-x-1$ do not have a $C$ relationship, find the range of real numbers for $a$;  
$(3)$ If the functions $f(x)=xe^{x}$ and $g(x)=m\sin x$ ($m \lt 0$) have a $C$ relationship on the interval $(0,\pi)$, find the range of real numbers for $m$.

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