In ΔABC, AB = 13, BC = 14, and AC = 15. Point X is chosen inside ΔABC such that 7AX + 5BX + 5CX is minimized. Let I be the incenter of ΔXBC. The quantity AI can be expressed in the form sqrt(a + b sqrt(c) - d) for positive integers a, b, c, d where c is not divisible by the square of any prime. What is a + b + c + d?