Coordinate Geometry: JEE Main 2024 Mathematics question

In a $\triangle A B C$, suppose $y=x$ is the equation of the bisector of the angle $B$ and the equation of the side $A C$ is $2 x-y=2$. If $2 A B=B C$ and the points $A$ and $B$ are respectively $(4,6)$ and $(\alpha, \beta)$, then $\alpha+2 \beta$ is equal to