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We construct an associated Deligne-Mumford stack $\\widetilde{\\mathcal{M}}$ with an induced semi-perfect obstruction theory of virtual dimension zero and define the generalized Donaldson-Thomas invariant of $W$ via Kirwan blowups to be the degree of the virtual cycle $[\\widetilde{\\mathcal{M}}]^{\\mathrm{vir}}$. 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