
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 \( A = \{1, 2, \cdots, n\} \) be a set, and let \( X \) and \( Y \) be non-empty subsets of \( A \) (allowing \( X \) to equal \( Y \)). Denote the largest element in \( X \) as \( \max X \) and the smallest element in \( Y \) as \( \min Y \). Find the number of ordered pairs of subsets \( (X, Y) \) such that \( \max X > \min Y \).

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