
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
Let  $\triangle ABC$  be a triangle with  $AB=85$ ,  $BC=125$ ,  $CA=140$ , and incircle  $\omega$ .  Let  $D$ ,  $E$ ,  $F$  be the points of tangency of  $\omega$  with  $\overline{BC}$ ,  $\overline{CA}$ ,  $\overline{AB}$  respectively, and furthermore denote by  $X$ ,  $Y$ ,  and  $Z$  the incenters of  $\triangle AEF$ ,  $\triangle BFD$ , and  $\triangle CDE$ , also respectively.  Find the circumradius of  $\triangle XYZ$ .

*Proposed by David Altizio*

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