We have a half-circle with endpoints $A$ and $B$ and center $S$. The points $C$ and $D$ lie on the half-circle such that $\angle BAC = 20^\circ$ and the lines $AC$ and $SD$ are perpendicular to each other. If the angle between the lines $AC$ and $BD$ is $x^\circ$, what is the value of $x$?

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