
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  $AB=20$  and  $AC=18$ .  $E$  is on segment  $AC$  and  $F$  is on segment  $AB$  such that  $AE=AF=8$ . Let  $BE$  and  $CF$  intersect at  $G$ . Given that  $AEGF$  is cyclic, then  $BC=m\sqrt{n}$  for positive integers  $m$  and  $n$  such that  $n$  is not divisible by the square of any prime. Compute  $100m+n$ .

*Proposed by James Lin*

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