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piece something out.
1. Lit. to add patches or pieces to something to make it complete. There is not quite enough cloth to make a shirt, but I think I can piece it out with some scraps of a complementary color for the collar. We managed to piece out the material that we needed.
2. Fig. to add missing parts to a story, explanation, or narrative to make it make sense. Before she passed out, she muttered a few things and we were able to piece the whole story out from that. We pieced out the story from the few bits we heard from her.