Gear ratios over X segments?

[fifthFunction]

how would you work out gear ratios?
say i want a 8:3 gear ratio spreed over 3 gears sections say like (2:1)2:3)2:1)=8:3 ratio
but no section can be 1:1 and no 2 numbers can be the same next to each other eg "(?:1)1:?)" as thy would cancel out and no number can be a floating point... i could be wrong about floating points tho but i feel like it will add an error to the reset location of the gears
so the only known number would be 8(input) 3(output) and 3(number of sections).... as far as i can tell the answer would be 4:6 but i have no clue how to get it using 8,3,3.
(2:1)2:3)2:1) * 3 = (8:4)4:6)6:3)

 

[fifthFunction]

did i post this in the right place?

 

[fifthFunction]

well i think i figured it out.... it cant seem to be done but i did work out a way using prime numbers to work out ever single combination of that can go it to an X:1 ratio then i just do the same for Y:i ratio then i just i just get all combinations of both the found X:Y ratios (with y being reversed) and remove duplicity numbers like (X:1)1:Y) the 1 would cancel out and there should never be any 1:1 things... then once all that is done i can make a list of the amount of gears needed and all veneration there of
tho other rule i am following is the ratios used in the gears can never be smaller than 1 or bigger than the in or output ratio number

everything seems to be working so far with no bugs in my code so hopefully everything will continue smoothly
here is a link to what i am making if any one wants to see
https://www.facebook.com/groups/236870953414442/