
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 the function $f(x) = \sqrt{\log_{3}(4x-1)} + \sqrt{16-2^{x}}$, its domain is A.  
(1) Find the set A;  
(2) If the function $g(x) = (\log_{2}x)^{2} - 2\log_{2}x - 1$, and $x \in A$, find the maximum and minimum values of the function $g(x)$ and the corresponding values of $x$.

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