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Tailor-made for massive connectivity and sporadic access, grant-free random access has become a promising candidate access protocol for massive machine-type communications (mMTC). Compared with conventional grant-based protocols, grant-free random access skips the exchange of scheduling information to reduce the signaling overhead, and facilitates sharing of access resources to enhance access efficiency. However, some challenges remain to be addressed in the receiver design, such as unknown identity of active users and multi-user interference (MUI) on shared access resources. In this work, we deal with the problem of joint user activity and data detection for grant-free random access. Specifically, the approximate message passing (AMP) algorithm is first employed to mitigate MUI and decouple the signals of different users. Then, we extend the data symbol alphabet to incorporate the null symbols from inactive users. In this way, the joint user activity and data detection problem is formulated as a clustering problem under the Gaussian mixture model. Furthermore, in conjunction with the AMP algorithm, a variational Bayesian inference based clustering (VBIC) algorithm is developed to solve this clustering problem. Simulation results show that, compared with state-of-art solutions, the proposed AMP-combined VBIC (AMP-VBIC) algorithm achieves a significant performance gain in detection accuracy.