
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 each non-empty subset \( A \) of \( S_{n} = \{1, 2, \cdots, n\} \), multiply each element \( k \) in \( A \) by \( (-1)^{k} \) and then sum the results. Find the total sum of all these sums.

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