A method is disclosed for operating an additive manufacturing system. The method may include wetting a reinforcement with a thermoset matrix inside of a print head, discharging the wetted reinforcement from the print head, and moving the print head in multiple directions during discharging. The method may also include exposing the thermoset matrix wetting the reinforcement to a cure energy to initiate a chemical reaction, and directing a medium onto the wetted reinforcement to regulate a rate of the chemical reaction.