Why couldn't a person in the OT be both priest and king?

In 1 Samuel 13, Saul is king and yet performs priestly duties even though he shouldn't have. Samuel rebukes Saul and tells him that his kingdom is slowly coming to an end because of his disobedience. My question is: where in the Bible is it established that a person could not hold both the positions of king and priest? Some other information that may or may not be interesting: 1) Melchizedek appears to Abram in Genesis 14 as BOTH priest and king. 2) Of course, Jesus fulfills both roles as the true Priest and true King. 3) The role of king of Israel wasn't established until the time of Saul and David, so during the writing of the Mosaic Law, kingship should not have been a thing. Were kings mentioned in the law? (if so, why?)