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In this paper, the cooperative output regulation problem for linear multi-agent systems with actuator faults is considered. It is assumed that the actuator faults are outage faults and loss-of-effectiveness faults. First, a distributed finite-time observer is designed to estimate the state of the exosystem. Based on the state of the finite-time observer, a distributed adaptive fault-tolerant controller is designed. Then, it is shown that the cooperative output regulation problem can be solved with the proposed fault-tolerant controller. Compared with the existing cooperative output regulation results, a novel lemma is introduced to guarantee the solvability of the regulator equations under actuator faults, and the developed controller is effective to compensate the actuator faults. Finally, a simulation example is presented to show the validity of the proposed method.