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Specifically, we work on not necessarily pseudoconvex domains $\\Omega\\subset\\mathbb{C}^n$ and define third and fourth order CR invariants on $\\bd\\Om$ and show that these invariants provide enough information to establish closed range for the $\\bar\\partial$-Laplacian in $L^2_{(0,q)}(\\Omega)$ for a given, fixed $q$. 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