
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 the domain of $f(x)$ is $(0, +\infty)$, and it satisfies $f(2) = 1$, $f(xy) = f(x) + f(y)$. Additionally, when $x_2 > x_1 > 0$, $f(x_2) > f(x_1)$.  
(1) Find the values of $f(1)$, $f(4)$, and $f(8)$.  
(2) If $f(x) + f(x - 2) \leq 3$ holds, find the range of $x$.

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