Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: $\bullet$ In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. $\bullet$ In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the $frst\textit{line. }$ $\bullet$ In the third line, Gugu 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 $frst\textit{line. }$ Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line. Please provide the sum of all possible values of $q$. 

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