
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 are positioned such that a smaller semicircle of diameter 2 sits at the top of a larger semicircle of diameter 4. A square is inscribed in the larger semicircle with its side length equal to the diameter of the smaller semicircle. Determine the area of the region inside the smaller semicircle and outside the larger semicircle, as well as outside the inscribed square. Express your answer in terms of \(\pi\) and simplest radical form.

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