
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 that $a \in \mathbb{R}$, the functions are defined as follows: $f(x) = \frac{a}{x} + \ln{x} - 1$ and $g(x) = (\ln{x} - 1)e^{x} + x$, where $e$ is the base of the natural logarithm.

1. Find the minimum value of the function $f(x)$ on the interval $(0, e]$.
2. Determine whether there exists a real number $x\_0 \in (0, e]$ such that the tangent line to the curve $y = g(x)$ at the point $x = x\_0$ is vertical to the $y$-axis. If it exists, find the value of $x\_0$; otherwise, explain the reason.

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