
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
Let $f(x) = \ln x + \frac{k}{x}, k \in \mathbb{R}$.

(I) If the tangent line to the curve $y = f(x)$ at the point $(e, f(e))$ is perpendicular to the line $x - 2 = 0$, find the interval where $f(x)$ is decreasing and the minimum value of $f(x)$. (Here, $e$ is the base of the natural logarithm.)

(II) If $f(x_1) - f(x_2) < x_1 - x_2$ holds for any $x_1 > x_2 > 0$, find the range of $k$.

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