TR-MO 2018 Problem 6

Initially, there are 2018 distinct boxes on a table. In the first stage, Yazan and Bozan, starting with Yazan, take turns make $2016$ moves each, such that, in each move, the person whose turn selects a pair of boxes that is not written on the board, and writes the pair on the board.

In the second stage, Bozan enumerates the $4032$ pairs with numbers from $1,2,\dots,4032$ , in whichever order he wants, and puts $k$ balls in each boxes written contained in the $k^{th}$ pair. Is there a strategy for Bozan that guarantees that the number of balls in each box are distinct?