
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
Six different colors are selected, and we need to paint the six faces of a cube, each in a unique color from the chosen ones. How many geometrically distinct ways can this be done? Geometrically distinct are two colorings that cannot be aligned with each other by rotating the cube around its center.

Solve the same problem for the case of coloring the faces of a regular dodecahedron with twelve different colors.

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