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I have $6$ integral variables, $m,z,p, m',z',p'$. I have a set of three inequalities:

$$m\leq m' \leq p+m$$ $$m \leq m'-z' \leq p+m$$ $$2p+2m+z \leq 2m'+p'\leq 2p+2m+z$$ (The last one is an equality).

Or in Matrix format

$ \begin{bmatrix} m \\ m \\ 2m+2p+z \\ \end{bmatrix} \leq \begin{bmatrix} 1 & 0 & 0 \\ 1 & -1 & 0 \\ 2 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} m'\\ z'\\ p'\\ \end{bmatrix} \leq \begin{bmatrix} m+p\\ m+p\\ 2m+2p+z\\ \end{bmatrix} $

I am trying to solve this system to bound $m',z',p'$ individually in terms of $m,z,p$. Meaning, I want an expression as: $ {\bf f}(m,z,p)\leq \begin{bmatrix} m' \\ z' \\ p' \\ \end{bmatrix}\leq {\bf g}(m,z,p) $ How can I invert this system?

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Alternatively, I want a change of basis from $m',z',p'$ to an orthogonal basis $l,m,n$ such that $ {\bf f}(m,z,p)\leq \begin{bmatrix} l \\ m \\ n \\ \end{bmatrix}\leq {\bf g}(m,z,p) $ again holds.

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