
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 $g(x) = \frac{1}{\cos\theta \cdot x} + \ln x$ is increasing on $[1, +\infty)$, and $\theta \in [0, \frac{\pi}{2})$, $f(x) = mx - \frac{m-1}{x} - \ln x$, where $m \in \mathbb{R}$.  
(1) Find the range of values for $\theta$;  
(2) If $h(x) = f(x) - g(x)$ is a monotonic function on $[1, +\infty)$, find the range of values for $m$;  
(3) If there exists at least one $x_0$ in $[1, e]$ such that $h(x_0) > \frac{2e}{x_0}$, find the range of values for $m$.

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