$A$ and $B$ are on a circle with radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen such that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$?

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