
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 $f(x) = x + \frac{\alpha}{x} + \ln x$ ($\alpha \in \mathbb{R}$),  
(1) Find the intervals of monotonicity and the extremum points of the function $f(x)$.  
(2) If for all $\alpha \in \left[\frac{1}{e}, 2e^2\right]$, the function $f(x)$ satisfies $f(x) < m$ for all $x \in [l, e]$, find the range of the real number $m$ (where $e$ is the base of the natural logarithm).

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