# A weird inequality

**Algebra**Level 5

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

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 \(\left\lfloor 1000M \right\rfloor\).