Invoking Complex Analysis!

Algebra Level 4

If xr=cos(π5r)+isin(π5r)x_r = \cos \left( \dfrac{\pi}{5^r} \right) + i \sin \left( \dfrac{\pi}{5^r} \right) , evaluate the value of:

(r=1xr)+(r=1xr)+r=1xr\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 (z)\Re(z), (z)\Im(z) and z|z| are the real part, imaginary part and absolute value of complex number zz.

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