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 \(a+b\)?