
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 $f(x) = mx - \frac{m-1+2e}{x} - \ln x$, where $m \in \mathbb{R}$, and the function $g(x) = \frac{1}{x\cos\theta} + \ln x$ is increasing on the interval $[1, +\infty)$, with $\theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

(I) Find the value of $\theta$;  
(II) Find the monotonic intervals and extremum of the function $f(x)$ when $m=0$;  
(III) If there exists at least one $x_0 \in [1, e]$ such that $f(x_0) \gt g(x_0)$, find the range of $m$.

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