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 \(A\times B\times C\).