A Unique Pair of Equal Chords

Let $C$ be a hyperbola with center $O$. Suppose there is exactly one pair of lines $A_1B_1$ and $A_2B_2$ through $O$, making an angle of $60^\circ$ with each other, such that $|A_1B_1|=|A_2B_2|$, where $A_1,B_1$ and $A_2,B_2$ are the intersection points of these lines with $C$. Find the range of the eccentricity of the hyperbola.