
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 \( B \) both be non-empty sets, such that \( A \cap B = \emptyset \) and \( A \cup B = \{1, 2, \ldots, n\} \) (where \( n \geq 3 \) and \( n \in \mathbb{N}^+ \)). Let the number of elements in \( A \) and \( B \) be \( a \) and \( b \) respectively. The number of ordered pairs \((A, B)\) that satisfy the condition “\( a \in B \)” and “\( b \in A \)” is denoted by \( a_n \). Find the expression for \( a_n \).

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