Consider a random walk on a circle of circumference n. The walker takes one unit step clockwise with probability p and one unit counterclockwise with probability q = 1 − p. Modify the program ErgodicChain to allow you to input n and p and compute the basic quantities for this chain. (a) For which values of n is this chain regular ergodic? (b) What is the limiting vector w? (c) Find the mean first passage matrix for n = 5 and p = .5. Verify that mij = d(n − d), where d is the clockwise distance from i to j.