Strike a chord

Consider concentric circles with respective radii π\pi and 2π2\pi. Choose two points on the larger circle independently and uniformly at random, and then join them to form a chord. If PP is the probability that this chord intersects the smaller circle at at least one point, then find 1000P\lfloor 1000P \rfloor.


Notation: \lfloor \cdot \rfloor denotes the floor function.

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