
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
In a given acute triangle  $\triangle ABC$  with the values of angles given (known as  $a$ ,  $b$ , and  $c$ ), the inscribed circle has points of tangency  $D,E,F$  where  $D$  is on  $BC$ ,  $E$  is on  $AB$ , and  $F$  is on  $AC$ . Circle  $\gamma$  has diameter  $BC$ , and intersects  $\overline{EF}$  at points  $X$  and  $Y$ . Find  $\tfrac{XY}{BC}$  in terms of the angles  $a$ ,  $b$ , and  $c$ .

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