
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 $f(x) = \lg [\log_{\frac{1}{2}} (\frac{1}{2}x - 1)]$, where the domain of the function is set $A$, and set $B = \{x | x < 1 \text{ or } x \geq 3\}$.

1. Find $A \cup B$ and $(R_B) \cap A$;
2. If $2^a \in A$ and $\log_2 (2a - 1) \in B$, find the range of values for the real number $a$.

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