Please answer this question by your own work. Note that I have checked the answers from other sources are not correct!!! Thank you

Problem 4. You play a game in which each turn you have a chance to win $2. If you have at least $2 then you win $2 with probability 1/3, and you lose $2 with probability 2/3. If you have $0 then you win $2 with probability 1/3, and nothing happens with probability 2/3. You start off with $4 and play forever. (a) (10p) Model this situation using a Markov chain, and show that there exists a unique stationary distribution for this Markov chain. (b) (10p) Determine the probability that you will have exactly $100 at least once during this game