
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
There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color, consistent on both sides. A larger equilateral triangle is constructed from six smaller paper triangles as shown. Three small triangles form an inner triangle, and three other triangles surround this inner triangle, all forming a larger equilateral triangle. Two such larger triangles are considered distinguishable if they cannot be superposed through any combination of translations, rotations, and/or reflections with matching colors on the small triangles. Given that there are eight different colors of triangles to choose from, how many distinguishable large equilateral triangles can be constructed?

[asy]
draw((0,0)--(10,0)--(5,8.7)--cycle);
draw((5,0)--(2.5,4.35)--(7.5,4.35)--cycle);
draw((3.75,2.17)--(6.25,2.17)--(5,5.2)--cycle);
[/asy]

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