
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
Let $f(x)=\ln x+\frac{k}{x}$, where $k\in \mathbb{R}$.  
$(1)$ If the tangent line of the curve $y=f(x)$ at the point $(e,f(e))$ is parallel to the line $y-2=0$, find the monotonicity and the minimum value of $f(x)$ (where $e$ is the base of the natural logarithm).  
$(2)$ If for any $x_{1} \gt x_{2} \gt 0$, $f(x_{1})-f(x_{2}) \lt x_{1}-x_{2}$ always holds, find the range of $k$.

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