
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, where each triangle is a solid color with the same color on both sides of the paper. You're tasked to construct a large equilateral triangle using nine of these small paper triangles arranged as shown. Three large triangles are considered distinguishable if it is not possible to match one onto the other, using translations, rotations, and/or reflections, aligning their small triangles of the same color. Given that there are now eight different colors available from which to choose, how many distinguishable large equilateral triangles can be constructed?

[asy]
draw((0,0)--(10,0)--(5,13)--cycle);
draw((5,0)--(2.5,4.33)--(7.5,4.33)--cycle);
draw((2.5,4.33)--(1.25,6.5)--(3.75,6.5)--cycle);
draw((7.5,4.33)--(6.25,6.5)--(8.75,6.5)--cycle);
draw((3.75,6.5)--(5,8.67)--(6.25,6.5)--cycle);
[/asy]

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