AM-GM + Complex!

Algebra Level 4

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 \(k\)?