
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
Let $A$ and $B$ be non-empty sets, and $A \cap B = \emptyset$, $A \cup B = \{1, 2, 3, \ldots, n\}$ ($n \geq 3$, $n \in \mathbb{N}^*$). Denote the number of elements in $A$ and $B$ as $a$ and $b$, respectively. The number of set pairs $(A, B)$ satisfying "$a \in B$ and $b \in A$" is denoted as $a_n$.
(1) Find the values of $a_3$ and $a_4$;
(2) Find $a_n$.

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