
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
Sixteen congruent equilateral triangles, each of a different color, are used to construct a regular tetrakis hexahedron (a polyhedron with 16 faces, each an equilateral triangle). You have only 12 colors available. How many distinguishable ways are there to color the tetrakis hexahedron? Note: Two colored tetrakis hexahedrons are distinguishable if neither can be rotated to look just like the other.

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