4 days to 2016 Tetrahdron problem

Geometry Level 5

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 \(m + n\).

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