
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 two sets, $A = \{x \mid 3 \leq 3^x \leq 27\}$ and $B = \{x \mid \log_{2}x > 1\}$:
1. Find $A \cap B$ and $(\complement_R B) \cup A$, where $\complement_R B$ denotes the complement of $B$ in the real numbers.
2. Let set $C$ be defined as $C = \{x \mid 1 < x < a\}$. If $C \subseteq A$, find the range of the real number $a$.

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