
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
Suppose  $\triangle ABC$  is such that  $AB=13$ ,  $AC=15$ , and  $BC=14$ .  It is given that there exists a unique point  $D$  on side  $\overline{BC}$  such that the Euler lines of  $\triangle ABD$  and  $\triangle ACD$  are parallel.  Determine the value of  $\tfrac{BD}{CD}$ .  (The  $\textit{Euler}$  line of a triangle  $ABC$  is the line connecting the centroid, circumcenter, and orthocenter of  $ABC$ .)

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