Given that \(a,b,c\) are non-negative real numbers satisfying \(abc=2\), then \[a\left(b^2+\dfrac{1}{b}\right)+b\left(c^2+\dfrac{1}{c}\right)+c\left(a^2+\dfrac{1}{a}\right)\ge k\] for some integer \(k\). What is \(k\)?

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