Did Mary actually leave Elizabeth before John was born?

36 Even Elizabeth your relative is going to have a child in her old age, and she who was said to be unable to conceive is in her sixth month. 37For no word from God will ever fail.”
38“I am the Lord’s servant,” Mary answered. “May your word to me be fulfilled.” Then the angel left her.
39At that time Mary got ready and hurried to a town in the hill country of Judea, 40where she entered Zechariah’s home and greeted Elizabeth.

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56Mary stayed with Elizabeth for about three months and then returned home.

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57 When it was time for Elizabeth to have her baby, she gave birth to a son.

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