# A weird inequality

Algebra Level 4

$\large{\frac { 2 }{ 3 } \left( { e }^{ \frac { x+y }{ 2 } }+{ e }^{ \frac { y+z }{ 2 } }+{ e }^{ \frac { z+x }{ 2 } } \right) -\frac { { e }^{ x }+{ e }^{ y }+{ e }^{ z } }{ 3 } }$

For some real numbers $$x,y,z$$ that satisfies the condition $$x+y+z=6$$ and the maximum value of above expression is $$M$$. Find $$\left\lfloor 1000M \right\rfloor$$.

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