
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
For a function $y=f(x)$ with domain $D$, if there exists an interval $\left[m,n\right]\subseteq D$ (where $m \lt n$), such that the function $y=f(x)$ simultaneously satisfies:<br/>① The function $y=f(x)$ is strictly increasing or strictly decreasing on $\left[m,n\right]$;<br/>② When the domain is $\left[m,n\right]$, the range of the function $y=f(x)$ is also $\left[m,n\right]$, then $\left[m,n\right]$ is called the "equal domain interval" of the function $y=f(x)$;<br/>$(1)$ If the interval $\left[1,2\right]$ is the "equal domain interval" of the function $y=a\cdot 2^{x}+b$ (where $a \gt 0$), find the values of the real numbers $a$ and $b$;<br/>$(2)$ Determine if the function $y=3-\frac{4}{x}$ has an "equal domain interval", and explain the reason;<br/>$(3)$ If the interval $\left[m,n\right]$ is an "equal domain interval" of the function $y=\frac{(a+1)x-\frac{1}{a}}{ax}$ (where $a\neq 0$), find the maximum value of $n-m$.

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