
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 with the same color on both sides. Construct a large equilateral triangle using four of these paper triangles. Two large triangles are considered distinguishable if it's not possible to place one on the other, using translations, rotations, and/or reflections, such that their corresponding small triangles match in color. You have eight different colors of triangles to choose from. Also, the top and bottom triangles in the larger triangle must be of the same color. How many distinguishable large equilateral triangles can be constructed?

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