
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
\( A_i \) (\( i = 1, 2, 3, 4 \)) are subsets of the set \( S = \{ 1, 2, \cdots, 2005 \} \). \( F \) is the set of all ordered quadruples \( (A_1, A_2, A_3, A_1) \). Find the value of \( \sum_{(A_1, A_2, A_3, A_4) \in F} |A_1 \cup A_2 \cup A_3 \cup A_4| \).

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