Circles \(\mathcal{C}_1, \mathcal{C}_2,\) and \(\mathcal{C}_3\) have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line \(t_1\) is a common internal tangent to \(\mathcal{C}_1\) and \(\mathcal{C}_2\) and has a positive slope, and line \(t_2\) is a common internal tangent to \(\mathcal{C}_2\) and \(\mathcal{C}_3\) and has a negative slope. Given that lines \(t_1\) and \(t_2\) intersect at \((x,y),\) and that \(x=p-q\sqrt{r},\) where \(p, q,\) and \(r\) are positive integers and \(r\) is not divisible by the square of any prime, find \(p+q+r.\)

(第二十四届AIME2 2006 第9题)