Performing motor tasks in virtual environments is best achieved with motion capture and animation of a 3D character that participants control in real time and perceive as being their avatar in the virtual environment. A strong Sense of Embodiment (SoE) for the virtual body not only relies on the feeling that the virtual body is their own (body ownership), but also that the virtual body moves in the world according to their will and replicates precisely their body movement (sense of agency). Within that frame of mind our specific aim is to demonstrate that the avatar can even be programmed to be better at executing a given task or to perform a movement that is normally difficult or impossible to execute precisely by the user. More specifically, our experimental task consists in asking subjects to follow with the hand a target that is animated using non-biological motion; the unnatural nature of the movement leads to systematic errors by the subjects. The challenge here is to introduce a subtle distortion between the position of the real hand and the position of the virtual hand, so that the virtual hand succeeds in performing the task while still letting subjects believe they are fully in control. Results of two experiments (N=16) show that our implementation of a distortion function, that we name the attraction well, successfully led participants to report being in control of the movement (agency) and being embodied in the avatar (body ownership) even when the distortion was above a threshold that they can detect. Furthermore, a progressive introduction of the distortion (starting without help, and introducing distortion on the go) could even further increase its acceptance.