
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

...

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system
A sector is cut out of a paper circle, as shown. The two resulting pieces of paper are "curled" to make two cones. The sum of the circumferences of the bases of these cones equals the circumference of the original paper circle. If the ratio of the lateral surface areas of the cones is \(2: 1\), what is the ratio of the volumes of the cones? 

(The lateral surface area of a cone is the area of the external surface not including the circular base. A cone with radius \(r\), height \(h\) and slant height \(s\) has lateral surface area equal to \(\pi r s\) and volume equal to \(\frac{1}{3} \pi r^{2} h\).)

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