HMMT 2017 U32

Let $a, b, c$ be non-negative real numbers such that $a b+b c+c a=3$. Suppose that

$$a^{3} b+b^{3} c+c^{3} a+2 a b c(a+b+c)=\frac{9}{2}$$

What is the minimum possible value of $a b^{3}+b c^{3}+c a^{3}$ ?