
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 $U = \mathbb{R}$, define the sets $A = \{x \mid 1 < x \leq 8\}$, $B = \{x \mid 2 < x < 9\}$, and $C = \{x \mid x \geq a\}$.
(1) Determine $A \cap B$ (the intersection of $A$ and $B$) and $A \cup B$ (the union of $A$ and $B$).
(2) If $A \cap C \neq \emptyset$ (not an empty set), find the range of values for $a$.

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