
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 $f(x)=\frac{{\ln x+a}}{x}$ and $g(x)=e^x+\frac{2}{x}-1$. <br/>$(1)$ If the tangent line of the graph of function $f(x)$ at $x=e$ is perpendicular to the line $2x-y+8=0$, find the extreme value of $f(x)$; <br/>$(2)$ For $x>0$, if $g(x)\geq f(x)$ always holds, find the range of real number $a$.

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