
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
For each positive integer \( n \) let \( S_{n} \) denote the set \( \{1,2,3, \ldots, n\} \). Compute the number of triples of subsets \( A, B, C \) of \( S_{2006} \) (not necessarily nonempty or proper) such that \( A \) is a subset of \( B \) and \( S_{2006}-A \) is a subset of \( C \).

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