
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 $n$ be an integer with $n \geq 3$, and let $P=\{1,2,3,\ldots,n\}$. Let $A$ and $B$ be two non-empty subsets of $P$. Let $a_n$ denote the number of pairs of sets $(A, B)$ such that the largest number in $A$ is less than the smallest number in $B$.
(1) Find $a_3$;
(2) Find $a_n$.

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