Let \(\mathbb{R}^2\) denote \(\mathbb{R} \times \mathbb{R}\). Let \(S=\left\{(a, b, c): a, b, c \in \mathbb{R} \text { and } a x^2+2 b x y+c y^2>0 \text { for all }(x, y) \in \mathbb{R}^2-\{(0,0)\}\right\} .\)
Then which of the following statements is (are) TRUE?

1. A \(\left(2, \frac{7}{2}, 6\right) \in S\)
2. B If \(\left(3, b, \frac{1}{12}\right) \in S\), then \(|2 b| < 1\).
3. C For any given \((a, b, c) \in S\), the system of linear equations
   \(\begin{aligned}& a x+b y=1 \\& b x+c y=-1\end{aligned}\)
   has a unique solution.
4. D For any given \((a, b, c) \in S\), the system of linear equations
   \(\begin{aligned}& (a+1) x+b y=0 \\& b x+(c+1) y=0\end{aligned}\)
   has a unique solution.