
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
Segment $AB$ has midpoint $M$, and segment $BM$ has midpoint $N$. Two semi-circles are constructed with $AB$ and $BM$ as diameters. Segment $MQ$ splits the region into two sections with the larger section twice the area of the smaller section. If the radius of the semi-circle on $BM$ is one-third that of the semi-circle on $AB$, find the degree measure of angle $AMQ$. Express your answer as a decimal to the nearest tenth.

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