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