Cubics

Algebra Level 4

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 $P(0)$ .

Cubics

This problem is from the OMO.