3d Geometry

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 ___________.