
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 pentagons share a common center, and their sides are parallel. The sides of two of the pentagons are 4 cm and 12 cm, respectively. The third pentagon divides the area between the first two pentagons in a ratio of \(1: 3\), starting from the smaller pentagon. Find the side length of the third pentagon.

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