Mathematics Question from JEE Main 2026 · 22 January, Shift 2

Let a vector $\vec a=\sqrt{2}\,\hat i-\hat j+\lambda \hat k,\ \lambda>0,$ make an obtuse angle with the vector $\vec b=-\lambda^2\hat i+4\sqrt{2}\hat j+4\sqrt{2}\hat k$ 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 _____.