A car travels due east at \frac{2}{3} mile per minute on a long, straight road. At the same time, a circular storm, whose radius is 51 miles, moves southeast at \frac{1}{2} \sqrt{2} mile per minute. At time t=0, the center of the storm is 110 miles due north of the car. At time t=t_{1} minutes, the car enters the storm circle, and at time t=t_{2} minutes, the car leaves the storm circle. Find \frac{1}{2}\left(t_{1}+t_{2}\right).