A circle of radius 1 has four circles $\omega_1, \omega_2, \omega_3$, and $\omega_4$ of equal radius internally tangent to it, such that $\omega_1$ is tangent to $\omega_2$, which is tangent to $\omega_3$, which is tangent to $\omega_4$, which is tangent to $\omega_1$. The radius of the circle externally tangent to $\omega_1, \omega_2, \omega_3$, and $\omega_4$ is $r$. If $r = a - \sqrt{b}$ for positive integers $a$ and $b$, compute $a + b$.

Let's think step by step and output the final answer within \boxed{}.
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