VMO 2013 — National examination, both days Question 7

Determine the number of ordered sextuples $(a,b,c,a',b',c')$ with all entries in $\{0,1,\ldots,14\}$ satisfying
$$\begin{cases}ab+a'b'\equiv1\pmod{15},\\ac+a'c'\equiv1\pmod{15},\\bc+b'c'\equiv1\pmod{15}.\end{cases}$$