Selection and Training Session

Let $q\in \mathbb{R}$. There are 10 distinct real numbers on the blackboard. Alex writes the following three lines of numbers:

1. In the first line Alex writes down every number of the form $a-b$, where $a,b$ are two (not necessarily distinct) numbers from the board;

2. In the second line Alex writes down every number of the form $qab$, where $a,b$ are two (not necessarily distinct) numbers from the first line;

3. In the third line Alex writes down every number of the form $a^2+b^2-c^2-d^2$, where $a,b,c,d$ are four (not necessarily distinct) numbers from the first line.

Determine all values of $q$ such that, regardless of the numbers on the board, every number in the second line is also a number in the third line.

(IMO-2017 Shortlist, Problem A2)