
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
Given the set \( I = \{1,2,3, \cdots, n\} \) where \( n \in \mathbb{N}^* \), select two non-empty subsets \( A \) and \( B \) of \( I \) such that the smallest number in \( B \) is greater than the largest number in \( A \). Denote the number of different selection methods as \( a_n \). It is clear that \( a_1 = 0 \) and \( a_2 = \binom{2}{2} = 1 \).

(1) Find \( a_n \).

(2) Let \( S_n \) be the sum of the first \( n \) terms of the sequence \( \{a_n\} \). Find \( S_n \).

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