
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  $AB=10$ ,  $AC=11$ , and circumradius  $6$ .  Points  $D$  and  $E$  are located on the circumcircle of  $\triangle ABC$  such that  $\triangle ADE$  is equilateral.  Line segments  $\overline{DE}$  and  $\overline{BC}$  intersect at  $X$ .  Find  $\tfrac{BX}{XC}$ .

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