4 days to 2016 Tetrahdron problem

In tetrahedron $SABC$ , the circumcircles of faces $SAB$ , $SBC$ , and $SCA$ each have radius 108.

The inscribed sphere of $SABC$ , centered at I, has radius 35. Additionally, $SI = 125$ . Let $R$ is the largest possible value of the circumradius of face $ABC$ .

Given that $R$ can be expressed in the form $\sqrt{ \dfrac{m}{n} }$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .