
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
Let \( S = \{1, 2, 3, \ldots, n\} \), and let \( A \) be a subset of \( S \). Arrange the elements of \( A \) in descending order, and then alternately subtract and add the successive elements starting from the largest to obtain a value called the alternating sum of \( A \). For example, if \( A = \{1, 4, 9, 6, 2\} \), rearrange it as \(\{9, 6, 4, 2, 1\}\), and its alternating sum is \( 9 - 6 + 4 - 2 + 1 = 6 \). Find the sum of all such alternating sums.

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