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I'm not the parent, but the ancient sumerian cuneiform script comes to mind.

Egyptian hieroglyphics did have a set of phonetic glyphs as well as the more complex logographic ones

Yes. Egyptian hieroglyphs have a full set of consonants, so you could write it that way. They just didn't because of historical spelling.

As you said Sumerian could be an example of totally "three step cue" system.