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Our main result is an asymptotic for the number of solutions to $\\pi_1 + \\pi_2 + \\pi_3 = f$ with $\\pi_1,\\pi_2,\\pi_3\\in S$, where $f:\\{1,\\dots,n\\}\\to G$ is an arbitary function satisfying $\\sum_{i=1}^n f(i) = \\sum G$. This extends recent work of Manners, Mrazovi\\'c, and the author. 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