
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
Define: If there exists a point in the graph of a function where the horizontal and vertical coordinates are equal, then this point is called an "equivalence point" of the graph of the function. For example, the point $(1,1)$ is an "equivalence point" of the graph of the function $y=2x-1$.  
$(1)$ Determine whether there exist "equivalence points" on the graphs of the functions $y=2x+1$ and $y=x^{2}-x+2$. If they exist, find the coordinates of the "equivalence points"; if not, explain the reason.  
$(2)$ Let the "equivalence points" of the graphs of the functions $y=\frac{9}{x}$ $(x>0)$ and $y=-x+b$ $(x>0)$ be points $A$ and $B$, respectively. Draw a perpendicular line from point $B$ to the x-axis, with the foot of the perpendicular being point $C$. If the area of $\triangle ABC$ is $3$, find the value of $b$.  
$(3)$ If the graph of the function $y=x^{2}-4$ for $x\geq m$ is denoted as $W_{1}$, and the graph obtained by folding $W_{1}$ along the line $x=m$ is denoted as $W_{2}$, and there are exactly two "equivalence points" on the combined graph of $W_{1}$ and $W_{2}$, determine the range of values for $m$.

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