
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
Triangle  $ABC$  has side lengths  $AB=13$ ,  $BC=14$ , and  $CA=15$ . Points  $D$  and  $E$  are chosen on  $AC$  and  $AB$ , respectively, such that quadrilateral  $BCDE$  is cyclic and when the triangle is folded along segment  $DE$ , point  $A$  lies on side  $BC$ . If the length of  $DE$  can be expressed as  $\tfrac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ , find  $100m+n$ .

*Proposd by Joseph Heerens*

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