
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: $2^x \leq 256$ and $\log_2{x} \geq \frac{1}{2}$,  
(1) Find the range of values for $x$;  
(2) Find the maximum and minimum values of the function $f(x) = \log_2\left(\frac{x}{2}\right) \cdot \log_{\sqrt{2}}\left(\frac{\sqrt{x}}{2}\right)$.

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