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Let's say that he dies in 1261 instead of 1259.

In 1259 and 1260, Hulagu Khan conquers Syria and Palestina. Latin states ends up vassalized.
In 1261, he invades and conquers Egypt before the death of Möngke Khan recalls him.

What then?

An advantage for the Ilkhans is that they now only have to deal with northern neighbours, the Golden Horde and Chagatai Khanate, giving them better chance of survival.
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