
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
A second square is formed with the diagonal of a square with side length 1 as its side length. A third square is formed with the diagonal of the second square as its side length, and so on, until the $n$-th square is formed. Let the area of the $n$-th square be $S_n$. By performing calculations and observing patterns, it can be conjectured that $S_n = \boxed{ }$.

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