AM-GM + Complex!

Algebra Level 5

If z4+3i1|z-4+3i| \leq 1 and α,β\alpha, \beta be the least and the greatest value of z|z| respectively and kk be the least value of x4+x2+4x\large \frac{x^4 + x^2 +4}{x} on the interval (0,)(0,\infty), then what is the value of kk?

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