
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