A cylindrical hole is drilled in a sphere x² + y² + z² = a² to make a bead. The hole is such that the center of the hole is drilled symmetrically through the center of the sphere. The width of the hole is determined so that when a string of length 2l (l > a) is passed through the hole, at least the half of the string remains out side the bead. What is the minimum width (diameter) of the hole that would allow this?
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