
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 $f(x) = \lg\left[\log_{\frac{1}{2}}\left(\frac{1}{2}x-1\right)\right]$ have its domain as set A, and set B = $\{x | x < 1 \text{ or } x \geq 3\}$.  
(1) Find $A \cup B$, $(\complement_{\mathbb{R}}B) \cap A$;  
(2) If $2^a \in A$, and $\log_2(2a-1) \in B$, find the range of the real number $a$.

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