Maximum Sum of Cubic Products

Algebra Level 5

Let \( x, y, z\) be non-negative real numbers satisfying the condition \( x+y+z = 1\). The maximum possible value of

\[ x^3y^3 + y^3z^3 + z^3x^3 \]

has the form \( \frac {a} {b} ,\) where \(a\) and \(b\) are positive, coprime integers. What is the value of \(a+b\)?

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