
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 triangular piece of paper is colored solid on both sides. When constructing a large equilateral triangle from these smaller triangles, two large triangles are considered distinguishable if it isn't possible to make them identical using any combination of movements including translations, rotations, or reflections. Eight different colors of triangles are available. How many distinguishable large equilateral triangles can be constructed?

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