
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 $f(x)=\dfrac{x}{\ln x}+ax$, where $x > 1$.

$(1)$ If $f(x)$ is monotonically decreasing on $(1,+\infty)$, find the range of the real number $a$;

$(2)$ If $a=2$, find the minimum value of the function $f(x)$;

$(3)$ If the equation $(2x-m)\ln x+x=0$ has two distinct real roots in $(1,e]$, find the range of the real number $m$.

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