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)