Invoking Complex Analysis!

Algebra Level 4

If \(x_r = \cos \left( \dfrac{\pi}{5^r} \right) + i \sin \left( \dfrac{\pi}{5^r} \right) \), evaluate the value of:

\[\large \Re\left(\prod_{r=1}^\infty x_r \right) + \Im\left(\prod_{r=1}^\infty x_r \right) + \left|\prod_{r=1}^\infty x_r \right| \]

where \(\Re(z)\), \(\Im(z)\) and \(|z|\) are the real part, imaginary part and absolute value of complex number \(z\).

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