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Say this theory. Ogedai doesn't die from his drunken stupor, and continues onward towards Europe

Southern Germany, Northern Italy, Central and Southern France, Poland, Balkans, Much of Lithuania, Austria, and maybe Benelux, will be ravaged by the Mongols.

Save for the following, from Bribing, Victory, or other reasons: Paris and the region surrounding it, Central Northern Germany, Teutonic Territory, Florence, Rome and South, Greece (Well just Athens and Thessalonica), Constantinople, and maybe a Balkan country, say Montenegro.

So they Pillage, Rape, Loot, etc.
Then they leave, and would probably have new Mongol states, or if the Khan dies, then every Mongol has to return to Mongolia to crown the new one.

What's left?
Florence survives so that means Dante can at least survive, Paris saved much of the Medieval knowledge, Rome still has the Pope, but much of Germany, France, and the Lower Regions have been ravaged, as well as Genoa and Venice, maybe Nicaea can take advantage of the ravaged Crusader states in Byzantium and restore the empire earlier, but whats left of Europe and its society?
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