Mathematics Question from JEE Main 2024 · 31 January, Shift 1

If $\alpha$ denotes the number of solutions of $|1 - i|^{x} = 2^{x}$ and $\beta = \dfrac{|z|}{\arg(z)}$, where

$$z = \frac{\pi}{4}(1 + i)^4 \left[\frac{1 - \sqrt{\pi}\, i}{\sqrt{\pi} + i} + \frac{\sqrt{\pi} - i}{1 + \sqrt{\pi}\, i}\right], \qquad i = \sqrt{-1},$$

then the distance of the point $(\alpha, \beta)$ from the line $4x - 3y = 7$ is: