
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $R$  be the largest possible value of the circumradius of face  $ABC$ . Given that  $R$  can be expressed in the form  $\sqrt{\frac{m}{n}}$ , where  $m$  and  $n$  are relatively prime positive integers, find  $m+n$ .

*Author: Alex Zhu*

Present the answer in LaTex format: \boxed{Your answer}
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