$(20$pts$)$ We have three students. Student$1$ belongs to section$1,$ where the midterm grades are distributed $N(88,3).$ Student$2$ belongs to section $2,$ where the midterm grades are distributed $N(85,5).$ Student$3$ belongs to section $2,$ where the midterm grades are distributed $N(80,4).$ Assume that the students' grades are independent of each other. What is the probability that the average grade of students $1$ and $2$ is more than the grade of student $3?$