AM-GM + Complex!

Algebra Level 5

If $|z-4+3i| \leq 1$ and $\alpha, \beta$ be the least and the greatest value of $|z|$ respectively and $k$ be the least value of $\large \frac{x^4 + x^2 +4}{x}$ on the interval $(0,\infty)$ , then what is the value of $k$ ?