A little card 'magic' - Oh no, it's Fibonacci again!

Browsing through some more Martin Gardner I stumbled across


The trick requires that you can false-shuffle effectively.
Being a numbers nerd, walking up the stations steps, 2 flights of 14 steps, I observe that taking either one or two steps at a time there are F14 (610) ways I can do it, with F1=1,F2=2,F3=3,F4=5 etc. A simple excel formula is =ROUND(phi^(n+1)/SQRT(5),0). With 28 steps it would be 514229 but the landing is too wide to be stepped over, with my little dachshund legs!
So if in my nerdish way I were to plan to execute all 610 different 1-2 step shuffles, how would I keep track of what I had done already? Let me illustrate with 8 steps. Possibilities are all the arrangements of 2222(1), 11222(10), 111122(15), 1111112(7), 11111111(1), 1+10+15+7+1=34=F8. Ok, that works - perhaps I might make it a resolution for 2021? Mmmm, perhaps not. That would be 610*2*28 = 34,160 steps on a dicky knee.