Maximizing a cyclic polynomial.

Algebra Level 5

Suppose \(a,\) \(b,\) and \(c\) are non-negative real numbers with \(a+b+c=1.\) The largest possible value of the expression \(ab^2+bc^2+ca^2\) can be written as \(\frac{n}{m},\) where \(n\) and \(m\) are coprime positive integers. What is the value of \(n+m\)?

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