Let $$w_1$$ and $$w_2$$ denote the circles $$x^2+y^2+10x-24y-87=0$$ and $$x^2 +y^2-10x-24y+153=0,$$ respectively. Let $$m$$ be the smallest positive value of $$a$$ for which the line $$y=ax$$ contains the center of a circle that is externally tangent to $$w_2$$ and internally tangent to $$w_1.$$ Given that $$m^2=\frac pq,$$ where $$p$$ and $$q$$ are relatively prime integers, find $$p+q.$$

(第二十三届AIME2 2005 第15题)