
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 $f(x) = e^x - e^{-x} - 2x$.

(I) Discuss the monotonicity of $f(x)$;

(II) Let $g(x) = f(2x) - 4bf(x)$. When $x > 0$, $g(x) > 0$, find the maximum value of $b$;

(III) Given that $1.4142 < \sqrt{2} < 1.4143$, estimate the approximate value of $\ln 2$ (accurate to 0.001).

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