Let \triangle A B C be a triangle with A B=13, B C=14, and C A=15. Let D, E, and F be the midpoints of A B, B C, and C A respectively. Imagine cutting \triangle A B C out of paper and then folding \triangle A F D up along F D, folding \triangle B E D up along D E, and folding \triangle C E F up along E F until A, B, and C coincide at a point G. The volume of the tetrahedron formed by vertices D, E, F, and G can be expressed as \frac{p \sqrt{q}}{r}, where p, q, and r are positive integers, p and r are relatively prime, and q is square-free. Find p+q+r.