Problem 24 (2025 AMC 8)

In trapezoid $ABCD$ with $AB \parallel CD$, angles $B$ and $C$ measure $60^\circ$, and $AB = DC$. The side lengths are all positive integers, and the perimeter of $ABCD$ is 30 units. How many non-congruent trapezoids satisfy all of these conditions? (Refer to AoPS page for diagram.)