
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 $f(x) = x - m \ln x - \frac{m-1}{x}$ (where $m \in \mathbb{R}$) and $g(x) = \frac{1}{2}x^{2} + e^{x} - xe^{x}$,
(1) Find the minimum value of $f(x)$ when $x \in [1, e]$,
(2) When $m \leqslant 2$, if there exists $x_{1} \in [e, e^{2}]$ such that for any $x_{2} \in [-2, 0]$, $f(x_{1}) \leqslant g(x_{2})$ holds, find the range of values for the real number $m$.

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