The hybrid affine system considered in the paper comes to existence when applying an impulse control of special form to the chain of four integrators. The goal of the control is to stabilize the system at the origin such that it approaches the equilibrium state along a given (desired) trajectory. The target trajectory is defined implicitly as that of the second-order integrator stabilized by means of a feedback in the form of nested saturators. The purpose of the study is to determine the range of the feedback coefficients for which this system is globally stable. The problem is shown to reduce to a simpler task of establishing stability of a second-order switched linear system with state-dependent switching law. It is proved that the latter system is stable for arbitrary switchings.