
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 $f(x) = 2(\log_{2}x)^{2} + 2a\log_{2}\frac{1}{x} + b$. It is known that when $x = \frac{1}{2}$, $f(x)$ has a minimum value of $-8$.

(1) Find the values of $a$ and $b$;  
(2) Find the set $A$ of $x$ values for which $f(x) > 0$.

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