Limited trigonometry

Calculus Level 5

$\large \sum_{r=1}^\infty \left[ \frac1{2^r} \tan\left( \frac{\pi /4}{2^r} \right) \right ]$

If the series above equals to $A\pi^B + C$ for integers $A,B$ and $C$ , find the value of $A\times B\times C$ .