Let the lines $L_1:\ \vec r=\hat i+2\hat j+3\hat k+\lambda(2\hat i+3\hat j+4\hat k)$ and $L_2:\ \vec r=(4\hat i+\hat j)+\mu(5\hat i+2\hat j+\hat k)$ intersect at point $R$. Let $P$ and $Q$ be points on $L_1$ and $L_2$ such that $|PR|=\sqrt{29}$ and $|OQ|=\sqrt{\frac{47}{3}}$. If $P$ lies in the first octant, then $27(OR)^2$ is equal to: