
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
(12 points) Let $f(x)$ be an increasing function defined on $(0, +\infty)$. For any $a, b \in (0, +\infty)$, it holds that $f(a \cdot b) = f(a) + f(b)$. Given that $f(2) = 1$, find:  
(1) The values of $f(1)$ and $f(4)$;  
(2) The solution set of the inequality $f(x^2) < 2f(4)$.

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