How to change 3d coordinate system?

Question

I want to change coordinate system from $(x,y,z)$ to $(x_n,y_n,z_n)$ (Figure is attached). What I mean is that I want to assign new $x,y,z$ axes.

What I know about new lines $(x_n,y_n,z_n)$:

1. origin is at $(x=k,y=0,z=k^2)$, where $k\geq1,$
2. $z_n$ line is parallel to $y$ line and passes through new origin $(x=k,y=0,z=k^2)$,
3. $y_n$ line is equal to line $z=(k+k^{-1})x-1$ where $y=0$,
4. $x_n$ line is equal to line $z=-\frac{1}{k+k^{-1}}x+\frac{k^4+2k^2}{k^2+1}$ where $y=0.$

How to perform such transformation and how to read points on old coordinate, for example (3,2,-1), on new coordinate system?