Its actually very simple. The cutting the deck parts actually do nothing (just do it yourself and see, you reassemble the cut when you put all the cards back together).
So once you reassmble the deck, you have 4 (that you took from the pile of 9 and placed at the bottom) + 10, then Ace #1, then 15 cards (because no matter how they cut, you are just putting them back together), then Ace #2, then 15 cards, then Ace #3, then 5 cards (what remains after you put 4 of the 9 to the bottom).
so
14, Ace, 15, Ace, 15, Ace, 5
Lets label all the cards starting at the top of the deck, that card will be card #1
1,2,3,4,5, 6 ACE, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 ACE, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38 ACE, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52
This will always be the order of the cards (if you aren't convinced the cut does nothing, try it be placing the Aces face up go through and cut however you want, put the deck together and count the cards between the aces.
So as you do the "up down, up down" starting at number 1, all odd numbers will be face up
then all odd number cards will be face up. Notice that all the Aces are even numbered. So we through away all the "up" cards, that is, all the odd numbers.
We are left with
2,4,6 ACE, 8, 10, 12, 14, 16, 18, 20, 22 ACE, 24, 26, 28, 30, 32, 34, 36, 38 ACE, 40, 42, 44, 46, 48, 50, 52
Except we have reversed the order. That is number 52 is now the top of the pile, and number 2 is the bottom
Doing the "up down, up down" again. The ups will be 52, 48, 44, 40, 36, 32, 28, 24, 20, 16, 12, 8, 4. The downs are then 50, 46, 42, 38 ACE, 34, 30, 26, 22 ACE, 18, 14, 10, 6 ACE, 2
Once again the order gets reversed and doing the "up down, up down" again, the ups will be
2, 10, 18, 26, 34, 42, 50. The downs are 6 ACE, 14, 22 ACE, 30, 38 ACE, 46.
Once again the order gets reversed and doing the "up down, up down" again the ups will be
46, 30, 14. The downs (the final three) are 38 ACE, 22 ACE, 6 ACE
And there you go, we have the three original cards.