How Do We Systematically Substitute Them?

Algebra Level 3

$P=\dfrac{a^2}{2a^2+bc}+\dfrac{b^2}{2b^2+ca}+\dfrac{c^2}{2c^2+ab}.$

Let $a,$ $b$ and $c$ be such that $a + b + c = 0$ .

Find the sum of all possible values of $P$ .