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Your challenge is to make the Byzantine empire last until the 16th century, and longer if you're really adventurous.
What re-forms, ASB's, butterflies or alliances would be nessisary to achieve this; and nothing like "a plauge wipes out every muslim in the world" and such.
The great problem was that you would have one great empourer who would conquer and do great things and then his son would be a pilock
Oh and a POD after 1070