In triangle A B C, angles A and B measure 60 degrees and 45 degrees, respectively. The bisector of angle A intersects \overline{B C} at T, and A T=24. The area of the triangle A B C can be written 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.