Pavan is shuffling cards. Given a deck of $4n$ cards $a_1, a_2, \dots, a_{4n}$, he will shuffle them into the order $$a_4, a_8, \dots, a_{4n}, a_3, a_7, \dots, a_{4n-1}, a_2, a_6, \dots, a_{4n-2}, a_1, a_5, \dots, a_{4n-3}$$ One day, Luke gifts Pavan a deck of cards labelled $1, 2, \dots, 4k$. Pavan applies his shuffle to the deck a finite number of times. He then realizes that he has shuffled the deck into perfectly reversed order: $4k, 4k-1, \dots, 1$.
Determine all possible values of $k$.