In circle $\omega$ with center $O$, let $\overline{AB}$ be a diameter of length 20 units. Point $C$ lies on the extension of $\overline{AB}$ through $A$ such that $AC = 10$ units. Point $T$ lies on $\omega$ such that line $CT$ is tangent to $\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose the distance from $B$ to $P$ is $x$ units. Find $x$. Let's think step by step and output the final answer within \boxed{}.
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