
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
Two semicircles A and B are inscribed inside a circle such that the diameters of each semicircle are chords of the circle that pass through its center. If the radius of semicircle A is \( r \) and the radius of semicircle B is \( 2r \), with the combined length of their diameters equal to the diameter of the circle, determine the ratio of the combined areas of the two semicircles to the area of the circle.

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