
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 quadrilateral \( A B C D \) be inscribed in a circle. The rays \( A B \) and \( D C \) intersect at point \( K \). It turns out that points \( B, D \), and the midpoints \( M \) and \( N \) 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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