Origami Probability

Geometry Level 4

A square sheet of paper has its center marked with point \(O\). A random point is uniformly chosen on the square paper, and this point is labeled \(P\). The paper is then folded so that point \(P\) coincides with point \(O\). Let the probability that a pentagon is formed after the fold be \(N\). Find the value of \(\lfloor 1000N \rfloor\).

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