Feet on the Asymptotes

Let $F_1, F_2$ be the foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. Drop perpendiculars from $P$ to the two asymptotes, with feet $D$ and $E$. If $\angle F_1PF_2 = 120^\circ$ and the area of $\triangle PF_1F_2$ equals $\sqrt{3}\,|PD|\cdot|PE|$, find the slopes of the two asymptotes.