
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
On a given circle, there are two fixed points \( A \) and \( B \) with \(\overrightarrow{AB} = \alpha\). An arbitrary circle passes through points \( A \) and \( B \). Through \( A \), a random line \( l \) is drawn, intersecting the circles again at points \( C \) and \( D \) (where \( C \) is a point on the given circle). Tangents to the circles at points \( C \) and \( D \) intersect at point \( M \). \( N \) is a point on line \( l \) such that \(|CN| = |AD| \) and \(|DN| = |CA| \). What values can the angle \( CMN \) take?

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