
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 \( \{1, 2, 3, \ldots, 8\} \) and each of its non-empty subsets, define a unique alternating sum as follows: arrange the numbers in the subset in decreasing order and alternately add and subtract successive numbers. For instance, the alternating sum for \( \{1, 3, 4, 7, 8\} \) would be \( 8-7+4-3+1=3 \) and for \( \{8\} \) it is \( 8 \). Find the sum of all such alternating sums for \( n=8 \).

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