Can an Eastern cleric be a pope? If so, what happens to the bishop and the Latin Church?

Eastern bishops can be cardinals, which means they have a direct impact on the papal conclave. We can assume that an Eastern priest can be pope, right? But if he is pope, he would have to take care of the diocese of Rome (the supreme pontiff's staff), but the diocese of Rome is of a different rite than his, so what does the new pope do? What happens to the Latin diocese? Can he simply bring the Eastern liturgy into the Basilica of St. John Lateran (for example)?

Eastern bishops can be cardinals, which means they have a direct impact on the papal conclave. We can assume that an Eastern priest can be pope, right? But if he is pope, he would have to take care of the diocese of Rome (the supreme pontiff's staff), but the diocese of Rome is of a different rite than his, so what does the new pope do? What happens to the Latin diocese? Can he simply bring the Eastern liturgy into the Basilica of St. John Lateran (for example)?