
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
Two regular polygons with perimeters \(a\) and \(b\) are circumscribed around a circle, while a third regular polygon is inscribed in this circle. The second and third polygons have twice as many sides as the first. Find the perimeter of the third polygon.

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