Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \dfrac{\pi}{3}$, $\arg(z_2) = \dfrac{\pi}{4}$ and $\arg(z_3) = \dfrac{\pi}{6}$. If $|z_1 z_2^2 + z_2 z_3^2 + z_3 z_1^2|^2 = a + b\sqrt{2}$, $a, b \in \mathbb{Z}$, then the value of $a^2 + b^2$ is: