
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
If the two roots of a quadratic equation differ by $1$, then the equation is called a "difference of $1$" equation. For example, $x^{2}+x=0$ is a "difference of $1" equation. <br/>$(1)$ Determine whether the following equations are "difference of $1" equations," and explain why: <br/>①$x^{2}-5x-6=0$; <br/>②$x^{2}-\sqrt{5}x+1=0$; <br/>$(2)$ Given that the equation concerning $x$, $x^{2}-(m-1)x-m=0$ ($m$ is a constant) is a "difference of $1" equation, find the value of $m$; <br/>$(3)$ If the equation concerning $x$, $ax^{2}+bx+1=0$ ($a$, $b$ are constants, $a \gt 0$) is a "difference of $1" equation, let $t=10a-b^{2}$, find the maximum value of $t$.

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