I would start with trigonometry.
Use Law of Cosines:
$|ED|^2 = 3^2+4^2-2\cdot 3\cdot 4 \cos(60^\circ) = 13$
So, $|ED| = \sqrt{13}$
Next, use Law of Sines:
$\dfrac{\sin 60^\circ}{\sqrt{13}} = \dfrac{\sin \left(\angle DEC\right)}{4}$
$\angle DEC = \arcsin\left(\dfrac{2 \sqrt{39}}{13}\right)$
That gives
$\angle ECF = 90^\circ-\arcsin\left(\dfrac{2 \sqrt{39}}{13}\right)$
$\angle AFC = 120^\circ - \angle CEF = 30^\circ+\arcsin\left(\dfrac{2 \sqrt{39}}{13}\right)$
Next, use Law of Sines again:
$\dfrac{\sin \left(\angle ECF\right)}{x} = \dfrac{\sin\left(\angle AFC \right)}{6}$
$\dfrac{\sqrt{13}}{13x} = \dfrac{\dfrac{1}{2}\dfrac{\sqrt{13}}{13}+\dfrac{ \sqrt{3} }{2}\dfrac{2\sqrt{39}}{13}}{6}$
$\dfrac{\cancel{\sqrt{13}}}{\cancel{13}x} = \dfrac{7\cancel{\sqrt{13}}}{2\cdot 6\cdot \cancel{13}}$
$x = \dfrac{12}{7}$