Oleksii wrote some \( 2n \) (\( n > 1 \)) consecutive positive integers on the board. After that, he grouped these numbers into pairs in some way, and within each pair, he multiplied the two numbers together. He then wrote the resulting \( n \) products on the board instead of the original numbers. Afterward, Anton wrote down the difference between the largest and the smallest of the numbers Oleksii wrote. Oleksii wants Anton to write the smallest possible number. What is the smallest number that can be written? Proposed by Oleksii Masalitin, Anton Trygub