Learn about packing shapes neatly into others, and the resulting geometric properties.

In the figure, each of the three circles is tangent to the other two and each side of the equilateral triangle is tangent to two of the circles.

If the length of one side of the triangle is 4, what is the radius of each circle?

**Details and assumptions**:

\(\bullet\) All the three circles have equal radii.

The area of the largest region can be written as \( A \pi \). What is \(A \)?

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