In triangle \(ABC\), angles \(A\) and \(B\) measure \(60\) degrees and \(45\) degrees, respectively. The bisector of angle \(A\) intersects \(\overline{BC}\) at \(T\), and \(AT=24\). The area of triangle \(ABC\) 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 \(a+b+c\).

(第十九届AIME1 2001 第4题)