Strike a chord

Discrete Mathematics Level 4

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

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

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