Let S_1 and S_2 be respectively the sets of all a ∈ R -{0} for which the system of linear equations a x+2 a y-3 a z=1; (2 a+1) x+(2 a+3) y+(a+1) z=2;…

Let $\displaystyle \mathrm{S}_1$ and $\displaystyle \mathrm{S}_2$ be respectively the sets of all $\displaystyle a \in \mathbb{R}-\{0\}$ for which the system of linear equations $$\begin{aligned} & a x+2 a y-3 a z=1 \\ & (2 a+1) x+(2 a+3) y+(a+1) z=2 \\ & (3 a+5) x+(a+5) y+(a+2) z=3 \end{aligned} $$ has unique solution and infinitely many solutions. Then