Let $a, b, c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that
$P(a) = b + c, P(b) = c + a, P(c) = a + b,$ and $P(a + b + c) = -16$, find $P(0)$.

This problem is from the OMO.

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