Matrices and Determinants: JEE Main 2023 Mathematics question

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