
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
Explore the rules and complete the related questions. Define the "$*$" operation: $(+2)*\left(+4\right)=+(2^{2}+4^{2})$; $(-4)*\left(-7\right)=[\left(-4\right)^{2}+\left(-7\right)^{2}]$; $(-2)*\left(+4\right)=-[\left(-2\right)^{2}+\left(+4\right)^{2}]$; $(+5)*\left(-7\right)=-[\left(+5\right)^{2}+\left(-7\right)^{2}]$; $0*\left(-5\right)=\left(-5\right)*0=\left(-5\right)^{2}$; $0*0=0^{2}+0^{2}=0$; $(+3)*0=0*\left(+3\right)=\left(+3\right)^{2}$. $(1)$ Calculate: ①$\left(-1\right)*\left(-1\right)$; ②$\left(-1\right)*\left[0*\left(-2\right)\right]$; $(2)$ Induct the rule of $*$ operation (in words or symbols): When two numbers are operated with $*$, ______; in particular, when $0$ is operated with any number, or any number is operated with $0$, ______; $(3)$ Does there exist integers $m$ and $n$ such that $\left(m-1\right)*\left(n+2\right)=-2$? If so, find the value of $m-n$; if not, explain the reason.

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