
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 \( S \) be the set of all \( n \)-tuples \(\left(X_{1}, \ldots, X_{n}\right)\), where \( X_{1}, \ldots, X_{n} \) are subsets of \(\{1,2, \ldots, 1000\}\), which do not all have to be different, and which can also be empty. For \( a=\left(X_{1}, \ldots, X_{n}\right) \in S \), denote

\[ E(a)=\text{number of elements of } X_{1} \cup \ldots \cup X_{n} \]

Find an explicit expression for the sum

\[ \sum_{a \in S} E(a) \]

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