
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
Given the function $f(x)=\frac{x}{{\ln x}}$.  
$(1)$ Find the monotonic intervals of $f(x)$ on $(1,+\infty)$.  
$(2)$ There exists $x_{0}\in (0,1)\cup (1,+\infty)$ such that $\frac{1}{{f(x_{0})}}\geq kx_{0}$. Find the range of real number $k$.  
$(3)$ If for all $m$ and $n\in [\sqrt{e},e^2]$, the inequality $\frac{f(m)-{f'}(n)}{a-2022}\leq 1$ always holds, find the range of real number $a$.

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