
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) Six different colors are chosen; it is required to paint the 6 faces of a cube, each in a unique color from the chosen ones. In how many geometrically distinct ways can this be done? Two colorings are called geometrically distinct if they cannot be made identical by rotating the cube around its center.

b) Solve the same problem for the case of painting the faces of a dodecahedron with 12 different colors.

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