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This letter presents an analysis of self-assembly of thin disk-shaped parts (diameter: 2 mm; thickness: 100 μm) with the objective of optimizing their angular alignment. The assembly proceeds continuously on a substrate that is pulled up through an air-water interface where thin parts with magnetic markers are floating. Angular deviations from the assembly site are significantly reduced by repositioning magnets that guide the self-assembly process. Temporary Faraday waves aid one-to-one part-to-site registration. Ninety parts are assembled, row by row, in 1 min. The assembly rate scales with the width of the assembly substrate. Compared with that of our previous work, the assembly rate is increased threefold due to reduced part-to-part interactions.