Mossy Easter Egg Sangaku

Geometry Level 5

Two large circles both with radius \(1\) pass through each other's center, as shown above.

The mid-sized circle is tangent to the two large circles at their centers.

The smallest circle is tangent to all the other 3 circles.

What is the area of the red-shaded region?

Give your answer to the nearest 3 decimal places.

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