
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 two sets \( A \) and \( B \), define
\[
A \Delta B = \{x \mid x \in A \text{ and } x \notin B\} \cup \{x \mid x \in B \text{ and } x \notin A\}.
\]

Let \( n > 1 \) be an integer, and let \( A_{1}, A_{2}, \ldots, A_{2n} \) be all the subsets of 
\[
S = \{1, 2, \cdots, n\}.
\]

Define \( M \) to be a \( 2^n \times 2^n \) table. For any \( i, j \in \{1, 2, \cdots, 2^n\} \), place the sum of all elements in the set \( A_i \Delta A_j \) (with the convention that the sum of elements in an empty set is 0) in the \(i\)-th row and \(j\)-th column of the table \( M \).

Find the sum of all the numbers in the table \( M \).

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