
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 $f(x) = mx - \frac{m-1}{x} - \ln x, m \in \mathbb{R}$. The function $g(x) = \frac{1}{x \cos \theta} + \ln x$ is increasing on $[1, +\infty)$, and $\theta \in \left[0, \frac{\pi}{2}\right)$.

(I) When $m=3$, find the equation of the tangent line to $f(x)$ at the point $P(1, f(1))$.

(II) Find the possible values of $\theta$.

(III) If $h(x) = f(x) - g(x)$ is monotonic on its domain, find the range of possible values for $m$.

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