
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
Point \( M \), located outside a circle with diameter \( AB \), is connected to points \( A \) and \( B \). Segments \( MA \) and \( MB \) intersect the circle at points \( C \) and \( D \) respectively. The area of the circle inscribed in triangle \( AMB \) is four times greater than the area of the circle inscribed in triangle \( CMD \). Find the angles of triangle \( AMB \), given that one of the angles is twice the size of another.

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