In triangle A B C, A B=10, B C=14, and C A=16. Let D be a point in the interior of \overline{B C}. Let I_{B} and I_{C} denote the incenters of triangles A B D and A C D, respectively. The circumcircles of triangles B I_{B} D and C I_{C} D meet at distinct points P and D. The maximum possible area of \triangle B P C can be expressed in the form a-b \sqrt{c}, where a, b and c are positive integers and c is not divisible by the square of any prime. Find a+b+c.