If $\alpha$ denotes the number of solutions of $|1-i|^x = 2^x$ and $\beta = \left( \frac{|z|}{\arg(z)} \right)$, where $z = \frac{\pi}{4}(1+i)^4 \left[ \frac{1-\sqrt{\pi}i}{\sqrt{\pi}+i} + \frac{\sqrt{\pi}-i}{1+\sqrt{\pi}i} \right]$, $i = \sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4x - 3y = 7$ is _________
