
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
Given the set $N=\{1, 2, 3, 4, \ldots, n\}$, where $A$ is a non-empty subset and $A \subseteq N$. Define the "alternating sum" of $A$ as follows: arrange the elements of set $A$ in descending order, then start from the largest number, alternately subtract and add the subsequent numbers until the last one. It is stipulated that the alternating sum of a single-element set is the element itself. For example, the alternating sum of the set $\{1, 2, 5, 7, 8\}$ is $8-7+5-2+1=5$, and the alternating sum of the set $\{4\}$ is $4$. When $n=2$, the non-empty subsets of $N=\{1, 2\}$ are $\{1\}$, $\{2\}$, $\{1, 2\}$, and the total sum of the alternating sums of these three sets is $S_2=1+2+(2-1)=4$. Then, when $n=3$, the total sum of the alternating sums of all non-empty subsets of $N=\{1, 2, 3\}$ is $S_3=\ $; the total sum of the alternating sums of all non-empty subsets of $N=\{1, 2, 3, 4, \ldots, n\}$ is $S_n=\ $.

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