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As you said, 行ってて is the colloquial contraction of 行っていて and we almost always use the contracted form in informal speech. There is a fairly big difference in meaning between 先に行って and 先に行ってて. 先に行って simply means "You go first." 先に行ってて means "You go first and wait for me/us." or "You go first and I/we will follow you."
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