Inequality

Let $x, y, z$ be positive real numbers satisfying $x(y^2 + z^2) = y + z$. Let $m$ be the minimum value of the expression $$P= \frac{1}{(1+x)^2} + \frac{1}{(1+y)^2} + \frac{1}{(1+z)^2} + \frac{4}{(1+x)(1+y)(1+z)}$$ If $m$ is in the form $\frac{\alpha}{\beta}$ for coprime integers $\alpha$ and $\beta$, compute $2\beta - \alpha$.