
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
Given that $D$ is a regular dodecahedron, and that $R$ is the rectangular solid (i.e., cuboid) centered at each face of $D$, with each cuboid having a horizontal length twice its depth and height equal to its depth, and the sum of all cuboids' volumes equaling half the volume of $D$. Find the ratio of the volume of $D$ to the combined volume of all the cuboids.

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