Let H : $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a>b>0$, be a hyperbola in the xy-plane whose conjugate axis LM subtends an angle of $60^{\circ}$ at one of its vertices $N$. Let the area of the triangle $L M N$ be $4 \sqrt{3}$.
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