
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 $f(x) = \ln x + \frac{a}{x} - 1$, where $a \in \mathbb{R}$.
(I) If the tangent to the curve $y=f(x)$ at the point $(1, f(1))$ is perpendicular to the line $x - y + 1 = 0$, find the extremum of the function.
(II) Let $g(x) = x + \frac{1}{x}$. When $a = -1$, if there exists an $x_0$ in the interval $[1, e]$ such that $g(x_0) < m[f(x_0) + 1]$, find the range of the real number $m$. (Here $e$ represents the natural logarithm base.)

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