
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 $f(x)=(a-1)\ln x+x+\frac{a}{x}$, where $a\in \mathbb{R}$.  
$(1)$ If $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(2,f(2))$.  
$(2)$ If for any $x\in (1,e]$, $f(x)-\frac{a}{x}>0$ holds, find the range of values for $a$.

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