Don't Get Confused!

$\begin{cases} \sqrt { \frac { 2 }{ z } } -\left( \frac { 1 }{ 2z } \sqrt { 8z } -\frac { z }{ 2 } \sqrt { 4z } \right) =\left\lfloor \sqrt { x } \right\rfloor \\ z\sqrt { z } =\left\lceil \sqrt { y } \right\rceil \end{cases}$

Given that $x$ and $y$ are integers satisfying the inequality $3106 \geq x > y> z$ and the system of equations above, where $z$ is a real number, find the maximum value of $x-y- \lfloor z \rfloor$ .

Notations :

• $\lfloor \cdot \rfloor$ denotes the floor function .

• $\lceil \cdot \rceil$ denotes the ceiling function .