
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 a function $f(x)$ with the domain $D: (-\infty, 0) \cup (0, +\infty)$, and it satisfies for any $x, y \in D$, $f(xy) = f(x) + f(y)$.

(I) Find the values of $f(1)$ and $f(-1)$;  
(II) Determine the parity of $f(x)$ and explain the reason;  
(III) If $f(4) = 1$, $f(3x+1) + f(2x-6) \leq 3$, and $f(x)$ is an increasing function on $(0, +\infty)$, find the range of $x$.

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