Mathematics Question from JEE Main 2023 · 11 April, Shift 2

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