A Random Card Shuffling Process

Consider a randomly shuffled deck of $2n$ cards with $n$ red cards and $n$ black cards. We study the average number of moves it takes to go from a randomly shuffled deck to a deck that alternates in color by performing the following move: If the top card and the bottom card of the deck differ in color place the top card at the bottom of the deck, otherwise, insert the top card randomly in the deck. We use tools from combinatorics, probability, and linear algebra to model this process as a finite Markov chain.