The position vectors of two \( 1 \, \text{kg} \) 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; \( \left( \alpha_1 = 1 \, \text{m} / \text{s}^2 , \alpha_2 = 3 n \, \text{m} / \text{s} , \alpha_3 = 2 \, \text{m} / \text{s} , \beta_1 = 2 \, \text{m} / \text{s} , \beta_2 = -1 \, \text{m} / \text{s}^2 , \beta_3 = 4 \rho \, \text{m} / \text{s} \right) \), where \( t \) is time, \( n \) and \( p \) are constants. At \( t = 1 \, \text{s} \), \( \left| \vec{V}_A \right| = \left| \vec{V}_B \right| \) 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 \( \sqrt{L} \, \text{kg} \, \text{m}^2 \, \text{s}^{-1} \). The value of \( L \) is ______.
