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 $z$ .