
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 the set $A = \{1, 2, 3, \cdots, n\}$ and its every non-empty subset, define a unique "alternating sum" as follows: rearrange the subset in decreasing order, and then starting from the largest number, alternately subtract or add the succeeding number to obtain the result. For example, the "alternating sum" of the set $\{1, 2, 4, 7, 10\}$ is $10-7+4-2+1=6$, the "alternating sum" of the set $\{7, 10\}$ is $10-7=3$, and the "alternating sum" of the set $\{5\}$ is $5$, etc. Find the total sum of the "alternating sums" of all subsets of the set $A$.

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