
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
Three regular polygons are inscribed in a circle, with the number of sides of each subsequent polygon being twice that of the previous one. The areas of the first two polygons are \( S_1 \) and \( S_2 \). Find the area of the third polygon.

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