
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
Synthesis and Exploration: If the one-variable quadratic equation about $x$, $ax^{2}+bx+c=0 (a\neq 0)$ has two real roots, and one root is greater than the other by $1$, then this type of equation is called a "neighboring root equation." For example, the one-variable quadratic equation $x^{2}+x=0$ has two roots $x_{1}=0$, $x_{2}=-1$, so the equation $x^{2}+x=0$ is a "neighboring root equation."  
$(1)$ By calculation, determine whether the following equations are "neighboring root equations":  
① $x^{2}+x-6=0$;  
② $2{x^2}-2\sqrt{5}x+2=0$.  
$(2)$ Given that the one-variable quadratic equation about $x$, $x^{2}-(m-2)x-2m=0 (m$ is a constant$)$ is a "neighboring root equation," find the value of $m$.

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