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I want to search for a flight with as many multi-day stops as possible, while keeping the price low. Whenever there is a stopover, I'd like it to be 24 hours plus, so I can explore that city.

Ie. give me the most circuitous route, so I can explore the world.

Yet nobody seems to have done this.

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You can do this sort of thing in ITA Matrix, it still takes a lot of work to craft the itinerary and learn the syntax, but I have used it to do exactly what you suggest, and found cheap 36 hour bonus stayovers.

I don't know how good is but you may want to look at airwander.com

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