Given a Pacman of radius $1$, and a mouth opening angle of $90^\circ$, what is the largest circular pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the radius be equal to $a\sqrt{b}+c$, where $b$ is square-free. Find $a+b+c$.

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