In circle $\omega$ with center $O$ and radius $r = 10$, points $A$ and $B$ lie on the circumference such that $AB = 12$. If the perpendicular from $O$ to $AB$ intersects $AB$ at point $P$, find the length of $OP$ and then compute the length of $AP + PB$. Let's think step by step and output the final answer within \boxed{}.
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