
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 exists a point on the graph of a function where the horizontal and vertical coordinates are equal, then that 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 $y=\frac{1}{2}x+\frac{1}{2}$.

$(1)$ Determine whether there exists an "equivalence point" on the graph of the function $y=\frac{1}{x-1}$ for $x>1$. If it exists, find the coordinates of the "equivalence point"; if not, explain the reason.

$(2)$ Given that the "equivalence points" of the graph of the function $y=x^{2}-2$ are the points $A(-1,-1)$ and $B(2,2)$.
① Given real numbers $m$ and $n$ satisfying $m^{2}-m-2=0$, $n^{2}-n-2=0$, and $m\neq n$, find the value of $m^{2}n+mn^{2}$.
② Given real numbers $p$ and $q$ satisfying $p^{2}=p+2$, $2q^{2}=q+1$, and $p\neq 2q$, find the value of $p^{2}+4q^{2}$.
③ If the graph of the function $y=x^{2}-2$ for $x\geqslant 1$ is denoted as $W_{1}$ and its reflection across the line $x=1$ is denoted as $W_{2}$, and the graph composed of $W_{1}$ and $W_{2}$ is denoted as $W$, find the "equivalence points" on the graph $W$.

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