
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  $AB=17$  and  $AC=23$ . Let  $G$  be the centroid of  $ABC$ , and let  $B_1$  and  $C_1$  be on the circumcircle of  $ABC$  with  $BB_1\parallel AC$  and  $CC_1\parallel AB$ . Given that  $G$  lies on  $B_1C_1$ , the value of  $BC^2$  can be expressed in the form  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Determine  $100m+n$ .

*Proposed by Michael Ren*

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