
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 $EF$ has midpoint $G$, and segment $FG$ has midpoint $H$. Semicircles are constructed with diameters $\overline{EF}$ and $\overline{FG}$ within the region shown. Segment $GQ$ is such that it divides the region into two sections, with the area of the section containing the center of the larger semicircle being twice the area of the other section. Determine the degree measure of angle $EGQ$.

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