Card Guessing

Consider a deck of cards. Each card has one of $$$4$$$ suits, and there are exactly $$$n$$$ cards for each suit — so, the total number of cards in the deck is $$$4n$$$. The deck is shuffled randomly so that each of $$$(4n)!$$$ possible orders of cards in the deck has the same probability of being the result of shuffling. Let $$$c_i$$$ be the $$$i$$$-th card of the deck (from top to bottom).

Monocarp starts drawing the cards from the deck one by one. Before drawing a card, he tries to guess its suit. Monocarp remembers the suits of the $$$k$$$ last cards, and his guess is the suit that appeared the least often among the last $$$k$$$ cards he has drawn. So, while drawing the $$$i$$$-th card, Monocarp guesses that its suit is the suit that appears the minimum number of times among the cards $$$c_{i-k}, c_{i-k+1}, \dots, c_{i-1}$$$ (if $$$i \le k$$$, Monocarp considers all previously drawn cards, that is, the cards $$$c_1, c_2, \dots, c_{i-1}$$$). If there are multiple suits that appeared the minimum number of times among the previous cards Monocarp remembers, he chooses a random suit out of those for his guess (all suits that appeared the minimum number of times have the same probability of being chosen).

After making a guess, Monocarp draws a card and compares its suit to his guess. If they match, then his guess was correct; otherwise it was incorrect.

Your task is to calculate the expected number of correct guesses Monocarp makes after drawing all $$$4n$$$ cards from the deck.