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This note revisits the problem of globally stabilizing an oscillator with a bounded delayed input. It establishes that an oscillator with input saturation can be globally asymptotically stabilized by a linear state feedback even in the presence of an arbitrarily large delay in the input. This result strengthens a recent result in the literature, which shows that such an oscillator can be globally asymptotically stabilized with a sufficiently small control input and the magnitude of the control input goes to zero as the delay increases to infinity. The controller constructed in this note thus improves the efficiency of the closed-loop system by fully utilizing the actuator capacity.