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The piecewise linear oscillator with a play considered in this work (Fig. 1 a), b)) was studied by Kleczka et al. in [4], where attention was focussed on sudden unexpected changes of the chaotic dynamics referred to as crisis. The global chaos of this oscillator was studied by Luo et al. [5], and an overview of its dynamics through codimension-1 bifurcation diagrams was presented by Wiercigroch [8]. Further, the analysed model is a simplified version to that for the analysis of gear-pair systems [3,6,7]. In the present work, the system behaviour was studied through direct numerical simulation and numerical continuation techniques, by means of our newly developed in-house Matlab-based computational suite, ABESPOL [1]. For a numerical continuation ABESPOL connects to the computational continuation core COCO [2]. Our resulting trajectories and Poincaré maps on the phase plane as well as resulting bifurcation diagrams (Fig. 1 c)) and basins of attraction show the influence of the system parameters on the dynamics. Moreover, interior and boundary crisis were observed and discussed, and parameter regions where various types of grazing incidence take place were detected and investigated further. The grazing induced bifurcations are also of our significant interest. We have understood from our earlier studies on a similar piecewise linear system that they have complex scenarios where there is an interplay between smooth bifurcations and grazing incidences.