Let $y = y(x)$ be the solution of the differential equation $\left(1 - x^2\right)dy = \left[ xy + \left(x^3 + 2\right)\sqrt{3\left(1 - x^2\right)} \right] dx$, $-1 < x < 1, y(0) = 0$. If $y\left(\frac{1}{2}\right) = \frac{m}{n}$, $m$ and $n$ are co-prime numbers, then $m + n$ is equal to _____.
