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Each stage consists of two coevolutionary populations, namely $${\\text {Pop}}_1$$<\/jats:tex-math>\n \n Pop<\/mml:mtext>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $${\\text {Pop}}_2$$<\/jats:tex-math>\n \n Pop<\/mml:mtext>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Throughout the three stages, $${\\text {Pop}}_1$$<\/jats:tex-math>\n \n Pop<\/mml:mtext>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is tasked with converging to the constrained Pareto front (CPF). However, $${\\text {Pop}}_2$$<\/jats:tex-math>\n \n Pop<\/mml:mtext>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is assigned with distinct tasks: (i) converging to the unconstrained Pareto front (UPF) in the push stage; (ii) utilizing constraint relaxation technique to discover the CPF in the pull stage; and (iii) revisiting the search for the UPF through knowledge transfer in the repush stage. Additionally, a novel reward-switching mechanism (RSM) is employed to transition between different stages, considering the extent of changes in the convergence and diversity of populations. Finally, the experimental results on three benchmark test sets and 30 real-world CMOPs demonstrate that TSRSM achieves competitive performance when compared with nine state-of-the-art CMOEAs. 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