
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
We are given a triangle  $ABC$ . Points  $D$  and  $E$  on the line  $AB$  are such that  $AD=AC$  and  $BE=BC$ , with the arrangment of points  $D - A - B - E$ . The circumscribed circles of the triangles  $DBC$  and  $EAC$  meet again at the point  $X\neq C$ , and the circumscribed circles of the triangles  $DEC$  and  $ABC$  meet again at the point  $Y\neq C$ . Find the measure of  $\angle ACB$  given the condition  $DY+EY=2XY$ .

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