
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 segment \( AB \) and on each of its halves, semicircles are constructed (all on the same side of \( AB \)). Given that the radius of the larger semicircle is \( R \), find the sum of the areas of the curvilinear triangles formed by constructing a circle tangent to all three given semicircles.

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