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There is currently no laser source that can provide both efficient excitation of mesospheric sodium and allow removal of the effects of guide-star elongation, as required for multi-conjugate adaptive optics in next-generation extremely large telescopes. We describe a Q-switched 1319 nm Nd:YAG laser that is mode-locked by injecting a short pulse into the Q-switched laser. It produces Q-switched mode-locked pulses that are synchronized to an external source. We also discuss how such pulses at 1064 nm and 1319 nm can be used to generate pulsed sodium-resonant guide stars via sum frequency generation, and thus satisfy the above requirements.