
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
When students at a certain school were studying folk paper-cutting art, they found that when cutting paper, they often fold the paper along a certain axis of symmetry. A rectangular paper with dimensions of $20dm \times 12dm$ can be folded once to obtain two shapes with dimensions of $10dm \times 12dm$ and $20dm \times 6dm$, respectively. The sum of their areas is $S_{1}=240dm^{2}$. Folding it twice can yield three shapes with dimensions of $5dm \times 12dm$, $10dm \times 6dm$, and $20dm \times 3dm$, and the sum of their areas is $S_{2}=180dm^{2}$. This pattern continues. The number of different shapes obtained by folding the paper four times is __; if the paper is folded $n$ times, then $\sum_{k=1}^{n}S_{k}=$____$dm^{2}$.

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