
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 $f(x) = e^x + ax$ and $g(x) = e^x \ln x$ (where $e$ is the base of the natural logarithm),
(1) If $f(x) > 0$ holds true for any $x \in \mathbb{R}$, determine the range of values for the negative real number $a$.
(2) When $a = -1$, does there exist an $x_0 \in (0, +\infty)$ such that the slope of the tangent line to the curve $C$: $y = g(x) - f(x)$ at the point $x = x_0$ is equal to the minimum value of $f(x)$ on $\mathbb{R}$? If such an $x_0$ exists, determine the number of such $x_0$; if not, explain why.

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