
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 $f(x)=\ln x+ax+\frac{a}{x}$.  
$(1)$ Discuss the monotonicity of the function $f(x)$.  
$(2)$ Investigate: Is there a real number $a$ such that the minimum value of the function $g(x)=f(x)-ax-2$ on the interval $(0,e^{2}]$ is $2$? If it exists, find the value of $a$; if it does not exist, please explain the reason.

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