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A non-orthogonal multiple access (NOMA)-inspired integrated sensing and communication (ISAC) framework is proposed, where a dual-functional base station (BS) transmits the composite communication and sensing signals. In contrast to treating the sensing signal as a harmful interference to communication, in this work, multiple beams of the sensing signal are exploited to convey extra information streams based on the concept of NOMA. Then, each communication user detects the extra information streams and the existing legacy information streams with the aid of successive interference cancellation (SIC). Based on the proposed framework, a multiple-objective optimization problem (MOOP) is formulated to characterize the trade-off between the communication throughput and sensing beampattern accuracy. For the general multiple-user scenario, the formulated MOOP is firstly converted to a single-objective optimization problem via the e-constraint method. Then, a double-layer block coordinate descent (BCD) algorithm is proposed by employing fractional programming and successive convex approximation to find a high-quality sub-optimal solution. For the special single-user scenario, the globally optimal solution can be obtained by transforming the MOOP into a convex quadratic semidefinite program. Moreover, it is rigorously proved that 1) in the multiple-user scenario, the proposed NOMA-inspired ISAC framework always outperforms the state-of-the-art sensing-interference-cancellation (SenIC) ISAC frameworks by further exploiting sensing signals for delivering extra information streams; 2) in the special single-user scenario, the proposed NOMA-inspired ISAC framework achieves the same performance as the existing SenIC ISAC frameworks, which reveals that the coordination of sensing interference is not necessarily required in this case. Numerical results verify the theoretical results.