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This paper studies a multi-antenna network integrated sensing and communication (ISAC) system, in which a set of multi-antenna base stations (BSs) employ the coordinated transmit beamforming to serve their respectively associated single-antenna communication users (CUs), and at the same time reuse the reflected information signals to perform joint target detection. In particular, we consider two target detection scenarios depending on the time synchronization among BSs. In Scenario \uppercase\expandafter{\romannumeral1}, these BSs are synchronized and can exploit the target-reflected signals over both the direct links (from each BS to target to itself) and the cross links (from each BS to target to other BSs) for joint detection. In Scenario \uppercase\expandafter{\romannumeral2}, these BSs are not synchronized and can only utilize target-reflected signals over the direct links for joint detection. For each scenario, we derive the detection probability under a specific false alarm probability at any given target location. Based on the derivation, we optimize the coordinated transmit beamforming at the BSs to maximize the minimum detection probability over a particular target area, while ensuring the minimum signal-to-interference-plus-noise ratio (SINR) constraints at the CUs, subject to the maximum transmit power constraints at the BSs. We use the semi-definite relaxation (SDR) technique to obtain highly-quality solutions to the formulated problems. Numerical results show that for each scenario, the proposed design achieves higher detection probability than the benchmark scheme based on communication design. It is also shown that the time synchronization among BSs is beneficial in enhancing the detection performance as more reflected signal paths are exploited.