
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 sets $A = \{x \mid 2 \leq 2^x \leq 8\}$ and $B = \{x \mid x > 2\}$, with the universal set $U = \mathbb{R}$.  
(1) Find $(\complement_U B) \cup A$.  
(2) Given the set $C = \{x \mid 1 < x < a\}$. If $C \subseteq A$, determine the range of the real number $a$.

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