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 $\lfloor 1000P \rfloor$.

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

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