
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  $ABC$  be a triangle with circumcenter  $O$  and let  $X$ ,  $Y$ ,  $Z$  be the midpoints of arcs  $BAC$ ,  $ABC$ ,  $ACB$  on its circumcircle. Let  $G$  and  $I$  denote the centroid of  $\triangle XYZ$  and the incenter of  $\triangle ABC$ .

Given that  $AB = 13$ ,  $BC = 14$ ,  $CA = 15$ , and  $\frac {GO}{GI} = \frac mn$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed by Evan Chen*

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