
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 $f(x)$ with domain $I$, if there exists an interval $\left[m,n\right]\subseteq I$ such that $f(x)$ is a monotonic function on the interval $\left[m,n\right]$, and the range of the function $y=f(x)$ for $x\in \left[m,n\right]$ is $\left[m,n\right]$, then the interval $\left[m,n\right]$ is called a "beautiful interval" of the function $f(x)$;  
$(1)$ Determine whether the functions $y=x^{2}$ ($x\in R$) and $y=3-\frac{4}{x}$ ($x \gt 0$) have a "beautiful interval". If they exist, write down one "beautiful interval" that satisfies the condition. (Provide the conclusion directly without requiring a proof)  
$(2)$ If $\left[m,n\right]$ is a "beautiful interval" of the function $f(x)=\frac{{({{a^2}+a})x-1}}{{{a^2}x}}$ ($a\neq 0$), find the maximum value of $n-m$.

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