Very bad idea for a very bad transmission:

Use a gokart forward and reverse gear box, then put the output into a retrodirect system, whether its chain based or a planetary gearbox. https://en.m.wikipedia.org/wiki/Retro-direct
They mentioned 2 ways of making a retrodirect system.the second way seems more practical since there'd be no crisscrossing chains. The Bikebarry double chain transmission could be considered to do this. It'd require changing the 2 larger sprockets to 2 different size freewheels.

Still there's much easier ways to build shifter bikes
 
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