
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 the integer $n \geq 3$, and the set $P = \{1, 2, 3, \ldots, n\}$. Let $A$ and $B$ be two non-empty subsets of $P$. Denote $a_n$ as the number of pairs of sets $(A, B)$ such that the maximum number in $A$ is less than the minimum number in $B$.  
(1) Find $a_3$;  
(2) Find $a_n$.

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