Let  $S_1$ and $S_2$ be respectively the sets of all $\mathrm{a}\in \mathrm{R}-\left\{0\right\}$ for which the system of linear equations
$\mathrm{ax}+2\mathrm{ay}-3\mathrm{az}=1$
$(2\mathrm{a}+1)\mathrm{x}+(2\mathrm{a}+3)\mathrm{y}+(\mathrm{a}+1)\mathrm{z}=2$
$(3\mathrm{a}+5)\mathrm{x}+(\mathrm{a}+5)\mathrm{y}+(\mathrm{a}+2)\mathrm{z}=3$
has unique solution and infinitely many solutions. Then