In each cell of a chessboard with $2$ rows and $2019$ columns, a real number is written such that:
1. There are no two numbers written in the first row that are equal to each other.
2. The numbers written in the second row coincide with (in some other order) the numbers written in the first row.
3. The two numbers written in each column are different, and they add up to a rational number.

Determine the maximum quantity of irrational numbers that can be in the chessboard.

Let's think step by step and output the final answer within \boxed{}.
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