
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
Definition: If there exist points on the graph of a function that are equidistant to the coordinate axes, then these points are called "equidistant points" of the graph of the function. For example, the points $(1,1)$ and $(-\frac{1}{3},\frac{1}{3})$ are "equidistant points" of the graph of the function $y=\frac{1}{2}x+\frac{1}{2}$.

$(1)$ Determine whether the graph of the function $y=x^{2}+2x$ has "equidistant points." If they exist, find the coordinates of the "equidistant points"; if they do not exist, explain the reason.

$(2)$ Let the "equidistant points" of the graph of the function $y=-\frac{4}{x}$ be $A$ and $B$, and the "equidistant points" of the graph of the function $y=-x+b$ be $C$. If the area of $\triangle ABC$ is $2\sqrt{3}$, find the expression of the function $y=-x+b$.

$(3)$ If the graph of the function $y=-x^{2}+(2+m)x+2m+2$ has exactly 2 "equidistant points," try to find the range of values for $m$.

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