Three Dimensional Geometry: JEE Main 2023 Mathematics question

Let the line $l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}$ meet the plane $P: x+2 y+3 z=4$ at the point $(\alpha, \beta, \gamma)$. If the angle between the line $l$ and the plane $P$ is $\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)$, then $\alpha+2 \beta+6 \gamma$ is equal to ___________.