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If xr=cos(π5r)+isin(π5r)x_r = \cos \left( \dfrac{\pi}{5^r} \right) + i \sin \left( \dfrac{\pi}{5^r} \right) xr=cos(5rπ)+isin(5rπ), evaluate the value of:
ℜ(∏r=1∞xr)+ℑ(∏r=1∞xr)+∣∏r=1∞xr∣\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| ℜ⎝⎛r=1∏∞xr⎠⎞+ℑ⎝⎛r=1∏∞xr⎠⎞+∣∣∣∣∣∣r=1∏∞xr∣∣∣∣∣∣
where ℜ(z)\Re(z)ℜ(z), ℑ(z)\Im(z)ℑ(z) and ∣z∣|z|∣z∣ are the real part, imaginary part and absolute value of complex number zzz.
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