As shown in the following figure, there is a line segment consisting of five line segments $AB, BC, CD, DE, and EA$ and $10$ intersection points of these five line segments. Find the number of ways to write $1$ or $2$ at each of the $10$ vertices so that the following conditions are satisfied. $\bigstar$ The sum of the four numbers written on each line segment $AB, BC, CD, DE, and EA$ is the same.