
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 $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{2, 4, 6, 8\}$ is $8-6+4-2=4$ and for $\{7\}$ it is simply $7$. However, if a subset contains exactly three elements, add an extra $3$ to the final sum. Find the sum of all such alternating sums for $n=8$.

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