The position vectors of two 1 kg particles, (A) and (B), are given by
\[
\vec{r}_A = (\alpha_1 t \hat{i} + \alpha_2 t^2 \hat{j} + \alpha_3 t^3 \hat{k}) \, \text{m}
\]
and
\[
\vec{r}_B = (\beta_1 t \hat{i} + \beta_2 t^2 \hat{j} + \beta_3 t^3 \hat{k}) \, \text{m}, \text{ respectively; }
\]
\[
(\alpha_1 = 1 \, \text{m/s}, \, \alpha_2 = 3 \, \text{m/s}^2, \, \alpha_3 = 2 \, \text{m/s}^3, \, \beta_1 = 2 \, \text{m/s}, \, \beta_2 = -1 \, \text{m/s}^2, \, \beta_3 = 4 \, \text{m/s}^3),
\]
where \( t \) is time, and \( n \) and \( p \) are constants. At \( t = 1 \, \text{s}, \, |\vec{V}_A| = |\vec{V}_B| \) and velocities \( \vec{V}_A \) and \( \vec{V}_B \) 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 \( \sqrt{L} \, \text{kgm}^2\text{s}^{-1} \). The value of \( L \) is ______.}
