Arithmetic mean

On a blackboard, there are 2016 integers, from 1 to 2016 (including 1 and 2016). You are allowed to remove two numbers you like, but you have to add the arithmetic mean as a new number. (If you delete for example 10 and 11, you have to write 10.5 as the new number). Therefore, the number of integers on the board decrease in every turn, and at the end of the game, only one integer will remain. What is the lowest value of this number?

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