
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 \( A_i (i = 1, 2, 3, 4) \) be subsets of the set \( S = \{1, 2, 3, \cdots, 2002\} \). Let \( F \) be the set of all quadruples \( (A_1, A_2, A_3, A_4) \). Find the value of \( \sum_{(A_1, A_2, A_3, A_4) \in \mathbf{F}} \left| A_1 \cup A_2 \cup A_3 \cup A_4 \right| \), where \(|x|\) denotes the number of elements in the set \( x \).

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