A large equilateral triangle is constructed from four smaller congruent equilateral triangles, each of a different solid color on both sides. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles can be constructed? Let's think step by step and output the final answer within \boxed{}.
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