
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 functions $f(x) = \ln(x)$ and $g(x) = \ln(x) - x + 2$.
1. Find the maximum value of function $g(x)$.
2. If the inequality $mf(x) \geq \frac{x-1}{x+1}$ holds for $x$ in the interval $[1, +\infty)$, determine the range of values for the real number $m$.
3. Given $\alpha \in \left(0, \frac{\pi}{2}\right)$, compare the values of $f(\tan(\alpha))$ and $-cos(2\alpha)$, and explain the reasons.

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