
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 universal set $U = \mathbb{R}$, set $A = \{x | \frac{1}{2} < 2^{x+1} < 8\}$, $B = \{x | 3a - 2 < x < 2a + 1\}$.  
$(1)$ When $a = 1$, find $A \cup (\overline{B})$;  
$(2)$ If $A \cap B = B$, find the range of values for $a$.

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