Maximization with constraints

Algebra Level 5

$x, y,$ and $z$ are real numbers such that $x+y+z= 0$ and $x^2 + y^2 + z^2 = 1$. The largest possible value of $x^2 y^2 z^2$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are positive coprime integers. What is the value of $a+b$?

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