A college professor teaches a certain course year after year. He has three favorite questions, and he always uses one of them in the final exam. He never uses the same question twice in a row. If he uses question A in one year, then the next year, he tosses a coin to choose between question B and C. If he uses question B, he tosses a pair of coins, and chooses question A if both coins show tails. Finally, if he uses question C, then he tosses three coins and uses question A if all of them show heads. Find the transition probability matrix for the resulting Markov chain.