Christmas Contest - Team Round - Problem 5

Let \bigtriangleup ABC be an equilateral triangle, and let O_1 and O_2 be its circumcircle and incircle, respectively.

If the radius of the O_1 is \frac{\sqrt{2}}{2} and the area inside \bigtriangleup ABC but outside O_2 is \frac{a\sqrt{b}}{c}-\frac{\pi}{d}, where a and c are relatively prime positive integers and b is not divisible by the square of any prime, compute a+b+c+d.