Universal Command Sequence
An \$n\$-maze is a \$n\times n\$ chessboard which has "walls" on some edges, and a "king" on the board that can move to the 4 adjacent cells, which cannot pass through any walls. Starting from any cell the king should be able to reach every cell on the board.
A command sequence is an array consisting of 4 distinct types of element (for example
[1,2,3,4,1,4,2,3,1,...]). Each type of element means a direction of the movement of the king. A command sequence can be "applied to" a maze, if the king can traverse every cell on the board by following the command sequence. For example a command sequence
[up,right,down] can be applied to a 2-maze that has no walls and the king is placed at the botton-left cell. If the king is going to pass through a wall or go outside the board, the command will be skipped.
For a given positive integer \$n\$, output a command sequence that can be applied to any \$n\$-maze. The existence of this sequence can be proved mathematically.See 1998 All-Russian Math Olympiad, Grade level 9, Day 1, Problem 4.
A positive integer
n. You can assume that
An array consisting of 4 distince types of elements.
Python 3 validator
Try it online. Test your generated sequence here. Usage tips can be found in the footer.
This is code-golf. Shortest code wins.