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Algebra Level pending

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

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 \(x-y- \lfloor z \rfloor \).

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