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For massive machine-type communications, centralized control may incur a prohibitively high overhead. Grant-free non-orthogonal multiple access (NOMA) provides possible solutions, yet poses new challenges for efficient receiver design. In this paper, we develop a joint user identification, channel estimation, and signal detection (JUICESD) algorithm. We divide the whole detection scheme into two modules: slot-wise multi-user detection (SMD) and combined signal and channel estimation (CSCE). SMD is designed to decouple the transmissions of different users by leveraging the approximate message passing (AMP) algorithms, and CSCE is designed to deal with the nonlinear coupling of activity state, channel coefficient and transmit signal of each user separately. To address the problem that the exact calculation of the messages exchanged within CSCE and between the two modules is complicated due to phase ambiguity issues, this paper proposes a rotationally invariant Gaussian mixture (RIGM) model, and develops an efficient JUICESD-RIGM algorithm. JUICESD-RIGM achieves a performance close to JUICESD with a much lower complexity. Capitalizing on the feature of RIGM, we further analyze the performance of JUICESD-RIGM with state evolution techniques. Numerical results demonstrate that the proposed algorithms achieve a significant performance improvement over the existing alternatives, and the derived state evolution method predicts the system performance accurately.