
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]


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Synthesis and Exploration: If a quadratic equation in $x$, $ax^{2}+bx+c=0 (a\neq 0)$ 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 quadratic equation $x^{2}+x=0$ has roots $x_{1}=0$ and $x_{2}=-1$, so the equation $x^{2}+x=0$ is a "neighboring root equation."
$(1)$ By calculation, determine whether the equation $2{x^2}-2\sqrt{5}x+2=0$ is a "neighboring root equation";
$(2)$ Given that the quadratic equation in $x$ $x^{2}-\left(m-2\right)x-2m=0 (m$ is a constant) is a "neighboring root equation," find the value of $m$.
$(3)$ If the quadratic equation in $x$ $ax^{2}+bx+2=0 (a, b$ are constants, and $a \lt 0)$ is a "neighboring root equation," let $t=2-b^{2}$, find the maximum value of $t$.

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