
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
Let the function $f(x)$ have a domain of $(0, +∞)$, and for any positive real numbers $x$, $y$, the property $f(xy) = f(x) + f(y)$ always holds. Given that $f(2) = 1$, and for $x > 1$, $f(x) > 0$,
(1) Find the value of $f\left(\frac{1}{2}\right)$;  
(2) Determine the monotonicity of $y = f(x)$ on $(0, +∞)$, and provide your proof;  
(3) Solve the inequality $f(x^2) > f(8x - 6) - 1$.

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