
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 $f(x) = e^x$, where $x \in \mathbb{R}$, and $g(x) = \ln x$, where $x \in (0, +\infty)$.

(I) If the line $y = kx + 2$ is tangent to the graph of $g(x)$, find the value of the real number $k$.

(II) Let $x > 0$, discuss the number of common points between the curve $y = f(x)$ and the curve $y = mx^2$ ($m > 0$).

(III) Let $a < b$, compare the size of $\frac{f(a) + f(b)}{2}$ and $\frac{f(b) - f(a)}{b - a}$, and explain the reason.

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