Cosines from sines

Circles $\omega_1$ , $\omega_2$ , and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$ , $P_2$ , and $P_3$ lie on $\omega_1$ , $\omega_2$ , and $\omega_3$ respectively such that $P_1 P_2 = P_2 P_3 = P_3 P_1$ and line $P_i P_{i+1}$ is tangent to $\omega_i$ for each $i = 1, 2, 3$ , where $P_4 = P_1$ . The area of $\triangle P_1 P_2 P_3$ can be written in the form $\sqrt{a} + \sqrt{b}$ for positive integers $a$ and $b$ . What is $a + b$ ?