
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 nonagons share a common center, and their sides are parallel. The sides of the nonagons are 8 cm and 56 cm. The area between the first two nonagons is divided by the third nonagon in the ratio $1:7$, measured from the smaller nonagon. Find the side length of the third nonagon. The answer is $8 \sqrt{7}$.

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