For a certain hyperbola
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,\]where $a > b,$ the angle between the asymptotes is $60^\circ.$  Find $\frac{a}{b}.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Let's think step by step and output the final answer within \boxed{}.
user