Dumping in public

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I don't know what Goemar used, but I personally use PGA whenever I make animations (for obvious reasons). I supose Animation Shop 3 would have some options somewhere, too, but PGA shows them on the main window right after opening the program: Color reduction and dither mode. These two options define how an image will be imported and how it's colors will be tweaked. PGA's default option (Optimal, Floyd Steinberg) tries to keep the original colors which works well as long as there is a maximum of 256 colors. As soon as you have more colors, they need to be merged.
I don't know why so many GIF animators have awkward default options.