Rotational Motion: JEE Main 2025 Physics question

The position vectors of two 1 kg particles, (A) and (B), are given by $\vec r_A=(\alpha_1t^2\hat i+\alpha_2t\hat j+\alpha_3t\hat k)$ m and $\vec r_B=(\beta_1t\hat i+\beta_2t^2\hat j+\beta_3t\hat k)$ m, respectively; ($\alpha_1=1$ m/s$^2$, $\alpha_2=3n$ m/s, $\alpha_3=2$ m/s, $\beta_1=2$ m/s, $\beta_2=-1$ m/s$^2$, $\beta_3=4p$ m/s), where t is time, n and p are constants. At $t=1$ s, $|\vec V_A|=|\vec V_B|$ and velocities $\vec V_A$ and $\vec V_B$ of the particles are orthogonal to each other. At $t=1$ s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{L}$ kg m$^2$s$^{-1}$. The value of L is ______.