\[ \sin \left( \displaystyle \prod_{n=1}^{\infty} \left[ \dfrac{(2n)^2}{(2n-1)(2n+1)}\right] \right) = \mathfrak{K} \]

Find \( \left \lfloor 10^3 \mathfrak{K} \right \rfloor \).

**Details and assumptions:**

- The argument of sine is given in radians.

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