This paper considers the four-state run-and-tumble particle model (RTP) at zero temperature. The model is an extension of the RTP model in one-dimension for two drift velocities, v = +/- v(0). This model is exactly solvable and imparts valuable insights for systems with finite temperature. However, at zero temperature, it yields uniform distributions for all parameter values and fails to provide any information about the structure of stationary distributions. To arrive at the model that more completely describes a zero temperature case, it is necessary to increase the number of discrete velocities. The four-state model with drifts v = +/- v(0), +/-gamma v(0) (where 0 <= gamma <= 1) is the simplest such an extension. In this paper, the four-state model at zero temperature is solved exactly and analyzed. The resulting stationary distributions indicate that fast particles accumulate at the walls and the slow particles are depleted. Taken all particles together, a dominant trend is accumulation, similar to what is observed for the two-state model for D > 0, however, reflecting a different physics behind it.