System's Twilight Hint Guide:

On the Way

Tell me what order I should get the balls in.

You should get the balls in this order:

1. purple
2. black
3. green
4. yellow
5. blue
6. purple
7. black
8. yellow
9. orange
10. red
11. purple
12. black
13. green
14. yellow
15. blue
16. orange (from bowl)
17. red
18. green
19. yellow

Could you show me a detailed, step-by-step solution?