
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 that $f(x)$ is an increasing function defined on the interval $(0, +\infty)$, with $f(2)=1$, and for any $a, b \in (0, +\infty)$, the equation $$f(a) - f(b) = f\left( \frac{a}{b} \right)$$ always holds.
(I) Find $f(8)$;
(II) Find the solution set for the inequality $$f(x+2) - f\left( \frac{1}{2x} \right) < 1 + f(x^2 + 4)$$.

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