
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 each non-empty subset of the set $N=\{1, 2, 3, \ldots, n\}$, define an "alternating sum" as follows: rearrange the subset in decreasing order, then start from the largest number and alternately subtract and add the succeeding numbers. For example, the alternating sum of the set $\{1, 2, 4, 6, 9\}$ is $9-6+4-2+1=6$, and the alternating sum of the set $\{5\}$ is $5$. When $n=2$ for the set $N=\{1, 2\}$, all non-empty subsets are $\{1\}$, $\{2\}$, $\{1, 2\}$, then the total sum of their "alternating sums" $S_2=1+2+(2-1)=4$. Please try to calculate the total sum of the "alternating sums" $S_3$, $S_4$ for $n=3$, $n=4$, and based on the results, guess the total sum of the "alternating sums" $S_n$ for each non-empty subset of the set $N=\{1, 2, 3, \ldots, n\}$.

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