Let \(\alpha\) be a root of the equation \((a-c) x^2+(b-a) x+\) \((c-b)=0\) where \(a, b, c\) are distinct real numbers …

Let \(\alpha\) be a root of the equation \((a-c) x^2+(b-a) x+\) \((c-b)=0\) where \(a, b, c\) are distinct real numbers such that the matrix \(\left[\begin{array}{ccc}\alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{array}\right]\) is singular. Then the value of \(\frac{(a-c)^2}{(b-a)(c-b)}+\frac{(b-a)^2}{(a-c)(c-b)}+\frac{(c-b)^2}{(a-c)(b-a)}\) is

[JEE Main 2023, 24 Jan (Shift 1)]