
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
Given a cyclic quadrilateral \(A B C D\), the rays \(A B\) and \(D C\) intersect at point \(K\). It is found that the points \(B\), \(D\), and the midpoints of segments \(A C\) and \(K C\) lie on the same circle. What values can the angle \(A D C\) take?

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