
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
Ten congruent equilateral triangles, each of a different color, are used to construct a regular icosahedron. How many distinguishable ways are there to construct the icosahedron, considering that two colored icosahedrons are distinguishable if neither can be rotated to look just like the other?

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