
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 three-stage launch vehicle consists of cylindrical stages. All these cylinders are similar to each other. The length of the middle stage is half the sum of the lengths of the first and third stages. When fueled, the mass of the middle stage is $13 / 6$ times less than the combined mass of the fueled first and third stages. Find the ratio of the lengths of the first and third stages. The masses of the engines, instruments, and the shells of the stages can be neglected.

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