
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 the set $\{1, 2, 3, \cdots, n\}$ and each of its non-empty subsets, we define the "alternating sum" as follows: arrange the numbers in each subset in descending order, then start from the largest number and alternately add and subtract each number (for example, the alternating sum of $\{1, 2, 4, 6, 9\}$ is $9-6+4-2+1=6$, and the alternating sum of $\{5\}$ is 5). For $n=7$, find the total sum of all these alternating sums.

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