
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
Suppose you have an unlimited supply of congruent equilateral triangles made of colored paper, with eight different colors available. Each triangle is colored solidly with the same color on both sides. You are tasked to construct a large equilateral triangle composed of four smaller triangles arranged as before. However, you additionally must ensure that at least one of the triangles is a specific color, say red. Two large triangles are considered distinguishable if it's impossible to superimpose one onto the other through any combination of translations, rotations, and reflections such that all corresponding small triangles match in color. How many distinguishable large equilateral triangles can you construct given these conditions?

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