Very skilled card handlers are capable of a technique whereby they cut a deck perfectly in half, then perfectly interleave the cards. If they start with a sorted deck and perform this technique flawlessly 52 times in a row, the deck will be restored to sorted order. Your challenge is to take
a deck of cards an integer array and determine whether it can be sorted using only Faro shuffles.
Mathematically, a Faro shuffle is a permutation on 2 n elements (for any positive integer n) which takes the element in position i (1-indexed) to position 2 i (mod 2 n+1). We would also like to be able to handle odd-length lists, so in that case, just add one element to the end of the list (a Joker, if you have one handy) and Faro shuffle the new list as above, but ignore the added dummy element when checking the list's order.
Write a program or function that takes a list of integers and returns or outputs a truthy if some number of Faro shuffles would cause that list to be sorted in nondescending order (even if that number is zero--small lists should give a truthy). Otherwise, return or output a falsy.
[1,1,2,3,5,8,13,21] => True [5,1,8,1,13,2,21,3] => True [9,36,5,34,2,10,1] => True [1,0] => True  => True  => True [3,2,1] => True [3,1,2] => False [9,8,7,6,5,4,3,2,1,0] => True [9,8,7,6,5,4,3,2,0,1] => False [3,1,4,1,5,9,2,6,9] => False [-1,-1,-1,-2] => True
This is code-golf so shortest source in bytes wins.