A random walk with reflecting barriers 0 and \(N\) is a Markov chain whose state space is \(E=\{0,1,2, \ldots, N\}\), which, at any state strictly between 0 and \(N\), moves next to either the state immediately to the left or immediately to the right with equal probability. If the chain is at state 0 , then it is certain to be at state 1 at the next time; and if it is at state \(N\), then it is certain to be at state \(N-1\) at the next time. For the random walk with reflecting barriers at 0 and 4 , find the conditional distribution of the state at time 6 , given that the initial state is each of: \(0,1,2,3\), and 4 .