Vector Algebra: JEE Main 2026 Mathematics question

Let a vector $\vec a=\sqrt2\hat i-\hat j+\lambda\hat k$, $\lambda>0$, make an obtuse angle with the vector $\vec b=-\lambda^2\hat i+4\sqrt2\hat j+4\sqrt2\hat k$ and an angle $\theta$, $\frac{\pi}{6}<\theta<\frac{\pi}{2}$, with the positive $z$-axis. If the set of all possible values of $\lambda$ is $(\alpha, \beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to ________.