Problem 9 (2005 AMC 8)

In quadrilateral $ABCD$, sides $\overline{AB}$ and $\overline{BC}$ both have length $10$, sides $\overline{CD}$ and $\overline{DA}$ both have length $17$, and the measure of angle $ADC$ is $60^{\circ}$. What is the length of diagonal $AC$? [See figure in original PDF]