HMMT 2022 U9

An $E$-shape is a geometric figure in the two-dimensional plane consisting of three rays pointing in the same direction, along with a line segment such that

• the endpoints of the rays all lie on the segment,
• the segment is perpendicular to all three rays,
• both endpoints of the segment are endpoints of rays.

Suppose two $E$-shapes intersect each other $N$ times in the plane for some positive integer $N$. Compute the maximum possible value of $N$.