Let \triangle A B C be an equilateral triangle. Points D, E, F are drawn on sides A B, B C, and C A respectively such that [A D F]=[B E D]+[C E F] and \triangle A D F \sim \triangle B E D \sim \triangle C E F. The ratio \frac{[A B C]}{[D E F]} can be expressed as \frac{a+b \sqrt{c}}{d}, where a, b, c, and d are positive integers such that a and d are relatively prime, and c is not divisible by the square of any prime. Find a+b+c+d.