
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
Given a set  $A$  which contains  $n$  elements. For any two distinct subsets  $A_{1}$ ,  $A_{2}$  of the given set  $A$ , we fix the number of elements of  $A_1 \cap A_2$ . Find the sum of all the numbers obtained in the described way.

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