
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 $S=\{1,2,3,\ldots,n\}$ ($n\in\mathbb{N}^*$, $n\geq2$), where $A$ and $B$ are two non-empty subsets of $S$, and it is given that the maximum number in set $A$ is less than the minimum number in set $B$. Denote the number of such pairs of sets $(A,B)$ that satisfy the condition as $P_n$.  
(1) Find the values of $P_2$ and $P_3$;  
(2) Find the expression for $P_n$.

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