\[\Large \displaystyle\int_{0}^{1} \left[ \dfrac{256.i^{6}.\dfrac{d^{2}}{d \theta^{2}} \left \{ \displaystyle\prod_{r=1}^{503} e^{(2r-1)i.\theta} \right \} }{ \displaystyle\prod_{r=0}^{502} e^{(2r+1)i.\theta} } \right]^{\dfrac{1}{4}} d\theta = ?\]

**Details and Assumptions**

Here \(e\) is the euler's number.

Here \(i=\sqrt{-1}\)

- \(\displaystyle{\prod_{n=1}^{k} n}\) means Continued Product of n upto \(k\)

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