
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 $N=\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets, define a type of summation called "alternating sum" as follows: for example, the alternating sum of the set $\{1, 2, 3, 4, 5\}$ is $5-4+3-2+1=3$, and the alternating sum of the set $\{3\}$ is $3$. When $n=2$ for the set $N=\{1, 2\}$, all non-empty subsets are $\{1\}, \{2\}, \{1, 2\}$, then the total of their "alternating sums" $S_2=1+2+(2-1)=4$. Please try to calculate the total of the "alternating sums" $S_3$ and $S_4$ for $n=3$ and $n=4$, respectively, and based on the calculation results, guess the total of the "alternating sums" $S_n$ for each non-empty subset of the set $N=\{1, 2, 3, \ldots, n\}$ equals to \_\_\_\_\_\_. (No need to provide proof)

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