$\DeclareMathOperator*{sinc}{sinc\,} \large \int_{0}^{\infty} \prod\limits_{k=0}^{n}\sinc\left(\frac{x}{2k+1}\right)dx=\, \frac{π}{2}$

Find the largest value of $n$ which satisfies the above equation.

• $n$ is a natural number.

• $\DeclareMathOperator*{sinc}{sinc\,} \sinc (x)=\frac{\sin x}{x}.$

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