$\alpha ,\beta$ be real numbers such that the system of linear equations $x+y+z=2, x+2y+3z=5, x+3y+\alpha z=\beta$ has infinitely many solutions. Let $\left|M\right|$ represent the determinant of the matrix $M=\left[\begin{array}{ccc}-2& -3+\beta & \sin \theta -2\\ 1& \sin \theta +2& \gamma \\ \alpha & 2\sin \theta -1& -\sin \theta +2+2\gamma \end{array}\right]$
Where $\gamma \in R$ and $\theta \in [0,2\pi ]$ and min value of determinant M is 8 and foot of perpendicular from point $P(\alpha -4,\beta -10,1)$ (where $(\alpha ,\beta )$ for which the above system of linear equation has infinitely many solution) to the plane $x+2y-2z-10=0$ is $F(a,b,c)$ then