
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
The domain of the function $f(x)$ is $D = \{x \mid x \neq 0\}$, and it satisfies the condition that for any $x_1, x_2 \in D$, we have $f(x_1 \cdot x_2) = f(x_1) + f(x_2)$.

(1) Find the value of $f(1)$;  
(2) Determine the parity (even or odd) of $f(x)$ and provide a proof;  
(3) Given that $f(4) = 1$, $f(3x + 1) + f(2x - 6) \leq 3$, and $f(x)$ is increasing on the interval $(0, +\infty)$, find the range of values for $x$.

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