
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 functions $f(x)=x-alnx$ and $g(x)=\frac{1+a}{x}$ where $a\in\mathbb{R}$.  
$(1)$ When $a=2e$, find the equation of the tangent line to the function $y=f(x)$ at the point $(e,f(e))$.  
$(2)$ Determine the intervals of monotonicity for the function $h(x)=f(x)+g(x)$.

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