150 Followers Problem

Algebra Level 5

$\large{\begin{cases} 2{ x }^{ 2 }-yz\left( 2-\sqrt{2}+\sqrt { 6 } \right) =\left( y-z \right) ^{ 2 }\times 2 \\ z^{ 2 }+xy(\sqrt { 3 } +2)={ (x+y) }^{ 2 } \\ \frac { y+z }{ x } =\frac { x-az }{ y-z } \end{cases}}$

For some positive real numbers $$x, y, z$$ satisfy the system of equations above. For some particular value of $$a$$ ( $$a \in \mathbb{R^{+}}$$ ) and the ratio $$\dfrac{z}{y}$$ ( let's say the ratio be $$m$$), there exist infinitely many solutions for the above given system in $$(x,y,z)$$. Find

$\left\lfloor 1000a \right\rfloor +\left\lceil 1000m \right\rceil$

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