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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