Now suppose you change your sensors so that at each time step t, they return the list of exact positions of the K cars, but the list of positions is shifted by a random number of indices (with wrap around). For example, if the true car positions at time step 1 are c11=(1,1),c12=(3,1),c13=(8,1),c14=(5,2), then e1 would be [(1,1),(3,1),(8,1),(5,2)], [(3,1),(8,1),(5,2),(1,1)], [(8,1),(5,2),(1,1),(3,1)], or [(5,2),(1,1),(3,1),(8,1)], each with probability 1/4. The shift can change from one timestep to the next. Define auxiliary variables z1,z2,…zt which can be used to model the relation between ct and et. Give an expression for p(ct∣ct−1) in terms of e1,⋯,et and zt. A description of auxiliary variables zt with their domains, followed by an expression for p(ct∣ct−1).