What is the solution for the arrow puzzle in room 9?

The guide is not helpful here, since the puzzle in the room apparently has changed in a recent release. Would be nice if you could update it!

The new riddle shows a board with 6x6 buttons, each of them showing an arrow:

rddlld

rrrddd

rddull

urulul

uulrlu

urruul

(r=right, l=left, u=up, d=down)

To solve it, you must push the buttons in the correct order. Each button vanishes after pushing it, and the next button you push must be one that the previous one pointed to.

This puzzle is hard (for those with math/computer science background: It's essentially Hamiltonian Path, which is NP-complete). I do not know if it can be solved elegantly (using the special structure of the board) or only by brute force. I decided to brute force it computationally. I used code from this stack overflow post: https://stackoverflow.com/questions/47982604/hamiltonian-path-using-python

One possible solution is (push the buttons in this order, starting at 1):

2 3 9 8 36 31

14 4 5 6 15 32

22 25 11 7 24 23

21 18 10 20 35 19

13 17 12 29 16 30

1 26 27 28 34 33

Fun facts: There are TONS of possible solutions (there are 144k ones that start with the same five buttons as this one alone), but they are hard to find since there are even way more possible paths (my code tried 400k possible paths until it found the first solution).

Please check the guide for Room 09 for the solutions to this rooms puzzles.