
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 functions $f(x) = x + \frac{1}{x}$ and $g(x) = a\ln{x} - \frac{a}{x}$.  
(1) Let the function $h(x) = f(x) - g(x)$, find the intervals of monotonicity for the function $h(x)$.  
(2) If there does not exist $x_0$ in the interval $[1, e]$ ($e = 2.71828\ldots$) such that $f(x_0) < g(x_0)$, find the range of the real number $a$.

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