HMMT November 2023 T10

Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0,0)$ and $(1,0)$, and the other two vertices are two distinct lattice points $(a, b),(c, d)$ with $0 \leq a, c \leq 59$ and $1 \leq b, d \leq 5$. (A concave quadrilateral is a quadrilateral with an angle strictly larger than $180^{\circ}$. A lattice point is a point with both coordinates integers.)