Given that $\angle BAC = 80^\circ$, $|AB| = |AC|$, $K \in [AB]$, $L \in [AB]$, $|AB|^2 = |AK| \cdot |AL|$, and $|BL| = |BC|$, find $\angle KCB$. If the answer 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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