Angular Momentum & its Conservation question from JEE Main 2025 (22 Jan, Shift 1)

The position vectors of two 1 kg particles, (A) and (B), are given by

$\vec{r}_A = \left(\alpha_1 t^2\hat{i} + \alpha_2 t\hat{j} + \alpha_3 t\hat{k}\right)\text{ m}$ and$\vec{r}_B = \left(\beta_1 t\hat{i} + \beta_2 t^2\hat{j} + \beta_3 t\hat{k}\right)\text{ m}$, respectively;

$\left(\alpha_1 = 1\text{ m/s}^2, \alpha_2 = 3n\text{ m/s}, \alpha_3 = 2\text{ m/s}, \beta_1 = 2\text{ m/s}, \beta_2 = -1\text{ m/s}^2, \beta_3 = 4p\text{ m/s}\right)$, where t is time, n and $p$ are constants. At and velocities $\vec{V}_A$ and $\vec{V}_B$ of the particles are orthogonal to each other. At $t = 1\text{ s}$, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{L}\text{ kg m}^{2}\text{ s}^{-1}$. The value of L is _________.